Motion Revision Notes

Speed is a measure of how fast an object moves. It is a scalar quantity, meaning it has magnitude only and no direction. Speed tells us the rate at which distance is covered over time.

In physics, speed is defined as the distance travelled divided by the time taken to travel that distance. This means if an object covers a certain distance in a given time, its speed is the ratio of these two quantities.

• Speed is scalar: it has magnitude but no direction.
• Speed = distance covered / time taken.
• It describes how fast an object moves, regardless of direction.

The formula to calculate speed is: v = s / t, where:

v is the speed (measured in metres per second, m/s),

s is the distance covered (measured in metres, m), and

• Speed (v) = Distance (s) ÷ Time (t)
• Distance (s) is measured in metres (m).
• Time (t) is measured in seconds (s).
• Speed (v) is measured in metres per second (m/s).

Suppose a car travels 150 metres in 10 seconds. To find its speed:

1. Write down the formula: v = s / t, where s = 150 m and t = 10 s.

2. Substitute the values: v = 150 m ÷ 10 s = 15 m/s.

• Speed = 150 m ÷ 10 s = 15 m/s
• The car’s speed is 15 metres per second.

Mixing units: Always ensure distance is in metres and time in seconds before calculating speed. If distance is given in kilometres or time in minutes, convert to metres and seconds respectively.

Forgetting to divide: Remember speed is distance divided by time, not multiplied.

Using average speed formula incorrectly: This objective focuses on speed for a specific time period, not average speed over multiple intervals.

• Convert units to metres and seconds before calculation.
• Use v = s / t correctly, do not multiply.
• Distinguish between speed and average speed concepts.

To measure speed practically, you can use a stopwatch and a measured distance. For example, mark a 20-metre track, time how long an object takes to travel this distance, then calculate speed using v = s / t.

Repeat measurements several times to improve accuracy and calculate an average time.

Ensure safety by keeping the area clear and using appropriate equipment.

• Measure a known distance accurately.
• Use a stopwatch to record time taken.
• Calculate speed using v = s / t.
• Repeat measurements to reduce errors.
• Maintain safety during experiments.

Velocity is defined as the speed of an object in a particular direction. It is a vector quantity, meaning it has both magnitude and direction.

Magnitude refers to how fast the object is moving (speed), while direction specifies the line along which the object moves.

For example, if a car travels at 50 km/h east, its velocity is 50 km/h east. If it changes direction but maintains the same speed, its velocity changes because direction is part of velocity.

• Velocity = speed with direction
• Vector quantity: has magnitude and direction
• Units typically metres per second (m/s) or kilometres per hour (km/h)

Speed is a scalar quantity that measures how fast an object moves regardless of direction. It only has magnitude.

Velocity is a vector quantity that includes both speed and direction.

Because velocity includes direction, two objects moving at the same speed but in different directions have different velocities.

• Speed: scalar, magnitude only
• Velocity: vector, magnitude and direction
• Velocity changes if speed or direction changes

Velocity is often represented as a vector arrow in diagrams. The length of the arrow shows the speed (magnitude), and the arrow points in the direction of motion.

In one-dimensional motion, velocity can be positive or negative depending on the chosen reference direction.

For example, motion to the right can be positive velocity, and motion to the left negative velocity.

• Vector arrow: length = speed, arrowhead = direction
• Positive and negative signs indicate direction in one dimension

Velocity can be calculated if displacement and time are known.

Displacement is the straight-line distance from the starting point to the final position, including direction.

The formula for average velocity is: v = Δx / Δt, where v is velocity, Δx is displacement, and Δt is time taken.

• Average velocity = displacement / time
• Displacement includes direction (vector)
• Units: metres per second (m/s)

If a runner completes a 100 m race in 10 seconds running straight east, average velocity = 100 m east / 10 s = 10 m/s east.

If the runner returns to the start in another 10 seconds, total displacement is zero, so average velocity over the whole time is zero, even though speed was not zero.

This shows velocity depends on direction and displacement, not just distance and speed.

• Velocity depends on displacement, not total distance
• Direction affects velocity value and sign
• Velocity can be zero if displacement is zero

Average speed is a scalar quantity that describes how fast an object moves over a certain distance, regardless of direction.

It is defined as the total distance travelled divided by the total time taken to travel that distance.

Average speed does not provide information about variations in speed during the journey; it only gives an overall rate of motion.

• Average speed is scalar (magnitude only).
• It is calculated over the entire journey.
• It ignores changes in speed or direction.

The formula to calculate average speed (v) is: v = total distance / total time.

Where: total distance (s) is measured in metres (m) or kilometres (km), and total time (t) is measured in seconds (s), minutes (min), or hours (h).

Units of average speed depend on the units used for distance and time, commonly metres per second (m/s) or kilometres per hour (km/h).

• v = s / t (average speed equals total distance divided by total time)
• Ensure distance and time units are compatible before calculation.

Example: A car travels 150 kilometres in 3 hours. Calculate its average speed.

Step 1: Identify total distance, s = 150 km; total time, t = 3 h.

Step 2: Use the formula v = s / t = 150 km / 3 h = 50 km/h (average speed).

• Write down known values clearly.
• Substitute values into the formula.
• Calculate and include units in the answer.

A frequent mistake is mixing units, such as using kilometres for distance and seconds for time without converting units.

Always convert time and distance to compatible units before calculating average speed.

Another error is confusing average speed with instantaneous speed; average speed is over the whole journey, not at a single moment.

• Convert units to be consistent (e.g., km and h, or m and s).
• Do not confuse average speed with velocity or instantaneous speed.
• Include units in the final answer to avoid ambiguity.

In experiments or real-life measurements, measure total distance accurately using a measuring tape, odometer, or map scale.

Measure total time using a stopwatch or timer, starting and stopping precisely when the motion begins and ends.

Repeat measurements to improve accuracy and calculate an average time if possible.

• Use appropriate measuring instruments for distance and time.
• Start and stop timing carefully to reduce errors.
• Repeat measurements and calculate averages to improve reliability.

A distance-time graph plots distance travelled (in metres, m) on the vertical (y) axis against time (in seconds, s) on the horizontal (x) axis.

The gradient (slope) of a distance-time graph represents the speed of the object. It is calculated as the change in distance divided by the change in time (gradient = Δdistance / Δtime).

A straight, sloping line indicates constant speed; the steeper the line, the greater the speed. A horizontal line shows the object is stationary (speed = 0). A curved line indicates changing speed (acceleration or deceleration).

• Y-axis: Distance (m)
• X-axis: Time (s)
• Gradient = speed (m/s) = Δdistance / Δtime
• Straight line slope: constant speed
• Horizontal line: object at rest
• Curved line: changing speed

A speed-time graph plots speed (in metres per second, m/s) on the vertical (y) axis against time (in seconds, s) on the horizontal (x) axis.

The gradient of a speed-time graph represents acceleration (m/s²), calculated as the change in speed divided by the change in time (gradient = Δspeed / Δtime).

A horizontal line indicates constant speed (zero acceleration). An upward sloping line shows acceleration, while a downward sloping line shows deceleration (negative acceleration).

• Y-axis: Speed (m/s)
• X-axis: Time (s)
• Gradient = acceleration (m/s²) = Δspeed / Δtime
• Horizontal line: constant speed
• Upward slope: acceleration
• Downward slope: deceleration

To calculate speed from a distance-time graph, select two points on a straight section of the graph.

Use the formula: speed = change in distance / change in time = (d₂ - d₁) / (t₂ - t₁).

Example: If an object moves from 20 m at 4 s to 50 m at 10 s, speed = (50 - 20) m / (10 - 4) s = 30 m / 6 s = 5 m/s.

• Identify two points on the graph
• Calculate Δdistance and Δtime
• Apply speed = Δdistance / Δtime
• Include units in the answer (m/s)
• Common error: mixing up axes or units

To calculate acceleration, choose two points on a straight section of the speed-time graph.

Use the formula: acceleration = change in speed / change in time = (v₂ - v₁) / (t₂ - t₁).

Example: If speed increases from 10 m/s at 2 s to 22 m/s at 6 s, acceleration = (22 - 10) m/s / (6 - 2) s = 12 m/s / 4 s = 3 m/s².

• Select two points on the graph
• Calculate Δspeed and Δtime
• Apply acceleration = Δspeed / Δtime
• Include units (m/s²)
• Common error: forgetting negative sign for deceleration

Distance-time graphs: a horizontal line means the object is stationary; a straight sloping line means constant speed; a curve means changing speed.

Speed-time graphs: a horizontal line means constant speed; an upward slope means acceleration; a downward slope means deceleration.

The area under a speed-time graph represents the distance travelled during that time interval (not required for this objective but useful context).

• Distance-time horizontal line = rest
• Distance-time straight slope = constant speed
• Distance-time curve = changing speed
• Speed-time horizontal line = constant speed
• Speed-time slope up = acceleration
• Speed-time slope down = deceleration

Motion graphs commonly plot distance or speed on the vertical (y) axis against time on the horizontal (x) axis. Distance is measured in metres (m), speed in metres per second (m/s), and time in seconds (s). Understanding which quantities are plotted is essential for correct interpretation.

In a distance-time graph, the vertical axis shows how far an object has travelled from a starting point at each time.

In a speed-time graph, the vertical axis shows the speed of the object at each moment in time.

• Distance-time graph: y-axis = distance (m), x-axis = time (s)
• Speed-time graph: y-axis = speed (m/s), x-axis = time (s)

When an object is at rest, its position does not change over time. On a distance-time graph, this is shown by a horizontal line because the distance remains constant as time passes.

If an object moves at constant speed, the distance increases uniformly over time. This appears as a straight line with a constant positive slope (gradient) on the distance-time graph. The gradient represents speed and is calculated as change in distance divided by change in time (speed = Δdistance / Δtime).

A steeper slope means a higher speed, while a shallower slope means a lower speed.

• Horizontal line = object at rest (speed = 0 m/s)
• Straight line with constant positive slope = constant speed
• Gradient of line = speed (m/s) = Δdistance (m) / Δtime (s)

Acceleration means the speed of the object is increasing over time. On a distance-time graph, this is shown by a curve that becomes steeper as time progresses, indicating the object covers more distance each second.

Deceleration means the speed is decreasing over time. On a distance-time graph, this appears as a curve that becomes less steep over time, showing the object covers less distance each second.

The curvature of the graph indicates changing speed: an upward curve (concave upwards) shows acceleration, a downward curve (concave downwards) shows deceleration.

• Curved line getting steeper = acceleration
• Curved line getting less steep = deceleration

In a speed-time graph, a horizontal line indicates constant speed because the speed value does not change with time.

An upward sloping straight line shows constant acceleration, as speed increases uniformly over time.

A downward sloping straight line shows constant deceleration, as speed decreases uniformly over time, eventually reaching zero if the line touches the time axis.

• Horizontal line = constant speed
• Straight line sloping upwards = constant acceleration
• Straight line sloping downwards = constant deceleration

By analysing the shape and gradient of distance-time and speed-time graphs, you can identify the motion state of an object.

Key points include: a horizontal line on a distance-time graph means rest; a straight line with constant slope means constant speed; curves indicate changing speed (acceleration or deceleration).

On speed-time graphs, the gradient represents acceleration, and the area under the graph represents distance travelled during the time interval.

• Distance-time graph: gradient = speed; shape indicates motion state
• Speed-time graph: gradient = acceleration; area under graph = distance
• Rest, constant speed, acceleration, and deceleration have distinct graph signatures

A distance-time graph plots the distance an object has travelled (vertical axis) against the time elapsed (horizontal axis).

The vertical axis is labelled 'Distance (m)' and the horizontal axis is labelled 'Time (s)', where metres (m) measure distance and seconds (s) measure time.

Each point on the graph represents the distance from the starting point at a particular time.

• Vertical axis: Distance (m)
• Horizontal axis: Time (s)

The slope (or gradient) of a straight line on a distance-time graph represents the speed of the object.

The slope is calculated as the change in vertical value divided by the change in horizontal value: slope = Δdistance / Δtime.

A steeper slope means a higher speed because the object covers more distance in less time, while a shallower slope means a lower speed.

• Slope = change in distance / change in time
• Steeper slope = higher speed
• Shallower slope = lower speed

To calculate speed from a straight section of the graph, select two points on the line.

Determine the coordinates of these points: (t₁, d₁) and (t₂, d₂), where t is time and d is distance.

Calculate the change in distance: Δd = d₂ - d₁ (in metres). Calculate the change in time: Δt = t₂ - t₁ (in seconds). Use the formula: speed = Δd / Δt, with units metres per second (m/s).

• Select two points on the straight line
• Calculate Δd = d₂ - d₁ (m)
• Calculate Δt = t₂ - t₁ (s)
• Speed = Δd / Δt (m/s)

Suppose a distance-time graph shows a straight line between points (2 s, 6 m) and (5 s, 15 m).

Calculate the change in distance: Δd = 15 m - 6 m = 9 m.

Calculate the change in time: Δt = 5 s - 2 s = 3 s. Calculate speed: speed = 9 m / 3 s = 3 m/s. The object moves at a constant speed of 3 metres per second during this interval.

• Δd = 15 m - 6 m = 9 m
• Δt = 5 s - 2 s = 3 s
• Speed = 9 m / 3 s = 3 m/s

Do not confuse the axes: distance must be on the vertical axis and time on the horizontal axis.

Always use two points on the straight section of the graph to find the slope; do not use points from curved sections as speed is not constant there.

Include units in every step of the calculation to avoid losing marks in exams.

• Check axis labels carefully
• Use points only on straight sections
• Show units in calculations

A speed-time graph plots speed (vertical axis) against time (horizontal axis). The speed is measured in metres per second (m/s), and time is measured in seconds (s).

The graph shows how an object's speed changes over time. The shape of the graph indicates whether the object is moving at constant speed, accelerating, or decelerating.

The gradient (slope) of a speed-time graph represents acceleration (change in speed per unit time). However, for this objective, the focus is on the area under the graph, which corresponds to the distance travelled.

• Vertical axis: Speed (m/s)
• Horizontal axis: Time (s)
• Gradient = acceleration (m/s²)
• Area under graph = distance travelled (m)

The key principle is that the area under a speed-time graph equals the distance travelled by the object during the time interval considered.

This is because distance = speed × time, and the area under the graph is calculated by multiplying speed (height) by time (base) for simple shapes.

For more complex shapes, the area can be split into rectangles, triangles, or trapezia, and their areas summed to find the total distance.

• Distance travelled = area under speed-time graph
• Area units: (m/s) × s = m (metres)

When an object moves at constant speed, the speed-time graph is a horizontal straight line.

The area under the graph is a rectangle with height equal to the constant speed and base equal to the time interval.

The distance travelled is calculated using the formula: Distance = speed × time.

• Graph shape: rectangle
• Formula: Distance = speed × time
• Example: Speed = 5 m/s, Time = 10 s, Distance = 5 × 10 = 50 m

When an object accelerates at a constant rate, the speed-time graph is a straight line with a positive or negative gradient.

The area under the graph forms a triangle or trapezium depending on the initial speed.

If starting from rest (initial speed zero), the area is a triangle: Area = ½ × base × height = ½ × time × final speed, which gives the distance travelled during acceleration.

• Graph shape: triangle or trapezium
• Triangle area formula: ½ × base × height
• Trapezium area formula: ½ × (sum of parallel sides) × height
• Example: Starting from rest, final speed = 8 m/s, time = 4 s, Distance = ½ × 4 × 8 = 16 m

An object accelerates uniformly from 0 m/s to 12 m/s in 6 seconds. Calculate the distance travelled during this time.

Step 1: Identify the shape under the graph: a triangle with base = 6 s and height = 12 m/s.

Step 2: Use the triangle area formula: Distance = ½ × base × height = ½ × 6 × 12 = 36 m.

• Distance = 36 metres
• Units: (m/s) × s = m
• Common error: forgetting to multiply by ½ for a triangle area

Acceleration due to gravity, symbolised as g, is the acceleration experienced by an object when it is in free fall near the Earth's surface, caused by the Earth's gravitational pull.

It is a vector quantity directed vertically downwards towards the centre of the Earth.

Acceleration means the rate of change of velocity with time, so g represents how quickly the velocity of a falling object increases each second due to gravity.

• g is the acceleration of an object due to Earth's gravity.
• It acts downwards, towards Earth's centre.
• It causes objects to speed up when falling freely.

Near the Earth's surface, the value of g is approximately constant and equal to 9.8 metres per second squared (9.8 m/s²).

This means that for every second an object falls freely, its velocity increases by about 9.8 m/s.

The unit m/s² means metres per second per second, indicating velocity change per unit time.

• g ≈ 9.8 m/s² near Earth's surface.
• Units of g are metres per second squared (m/s²).
• This value is used in most calculations involving free fall.

The acceleration due to gravity arises from the gravitational force exerted by the Earth on objects near its surface.

Although g varies slightly with altitude and location (due to Earth's shape and rotation), it is treated as constant at 9.8 m/s² for most practical physics problems.

This simplification allows straightforward calculations of motion under gravity without complex corrections.

• g results from Earth's gravitational force.
• g varies slightly but is approximated as constant at 9.8 m/s².
• This approximation simplifies calculations.

When an object is dropped (neglecting air resistance), it accelerates downwards at 9.8 m/s².

This means its velocity increases by 9.8 m/s every second it falls.

The displacement s of the object after time t can be calculated using equations of motion with constant acceleration g, for example: s = ½ g t², if starting from rest.

• Objects accelerate downwards at 9.8 m/s² when in free fall.
• Velocity increases by 9.8 m/s every second.
• Displacement under free fall can be calculated using s = ½ g t².

Example: Calculate the velocity of a ball dropped from rest after 3 seconds.

Formula: v = g × t (since initial velocity u = 0 and acceleration a = g)

Substitute: v = 9.8 m/s² × 3 s = 29.4 m/s (downwards)

• Formula: v = g × t
• Substitution: v = 9.8 × 3 = 29.4 m/s
• Answer: Velocity after 3 s is 29.4 m/s downwards.
• Common error: forgetting units or mixing up acceleration and velocity.

Acceleration is defined as the rate at which an object's velocity changes with time. Since velocity includes both speed and direction, acceleration can occur when an object speeds up, slows down, or changes direction.

Mathematically, acceleration (a) is given by the formula:

\[ a = \frac{\Delta v}{\Delta t} \] where \( \Delta v \) is the change in velocity and \( \Delta t \) is the time interval over which this change occurs.

• Acceleration is a vector quantity, meaning it has both magnitude and direction.
• Velocity change (\( \Delta v \)) can be positive (speeding up) or negative (slowing down).
• If direction changes but speed remains constant, acceleration is still present.

Velocity is measured in metres per second (m/s), and time in seconds (s). Therefore, acceleration units are metres per second squared (m/s²).

Because acceleration is a vector, its direction is the same as the direction of the change in velocity. This means acceleration can be positive or negative depending on the chosen reference direction.

Negative acceleration is often called deceleration, indicating a reduction in speed along the chosen direction.

• Units of acceleration: m/s².
• Acceleration direction depends on velocity change direction.
• Deceleration is acceleration opposite to velocity direction.

To calculate acceleration, determine the initial velocity (\( v_i \)) and final velocity (\( v_f \)) of the object, then find the change in velocity:

\[ \Delta v = v_f - v_i \] Next, measure the time interval (\( \Delta t \)) over which this change occurs.

Substitute these values into the formula: \[ a = \frac{v_f - v_i}{\Delta t} \] and calculate the acceleration value, including its sign and units.

• Identify initial and final velocities with direction.
• Calculate \( \Delta v = v_f - v_i \).
• Measure time interval \( \Delta t \).
• Substitute into \( a = \Delta v / \Delta t \).
• Include units (m/s²) and sign for direction.

An object initially moves at 5.0 m/s to the right and accelerates to 15.0 m/s to the right in 4.0 seconds. Calculate its acceleration.

Step 1: Identify initial velocity \( v_i = +5.0 \, \mathrm{m/s} \) (right is positive).

Step 2: Identify final velocity \( v_f = +15.0 \, \mathrm{m/s} \). Step 3: Calculate change in velocity: \( \Delta v = 15.0 - 5.0 = +10.0 \, \mathrm{m/s} \). Step 4: Time interval \( \Delta t = 4.0 \, \mathrm{s} \). Step 5: Calculate acceleration: \[ a = \frac{10.0 \, \mathrm{m/s}}{4.0 \, \mathrm{s}} = 2.5 \, \mathrm{m/s}^2 \] The positive sign indicates acceleration in the same direction as velocity (speeding up).

• Write down known values with units and directions.
• Calculate \( \Delta v \) carefully, considering direction.
• Divide by \( \Delta t \) to find acceleration.
• Include units and interpret sign.

A frequent mistake is ignoring the vector nature of velocity and acceleration, leading to incorrect signs in calculations.

Another error is mixing units or forgetting to include units in the final answer.

Sometimes, students confuse speed and velocity; acceleration depends on velocity changes, not just speed changes.

• Always consider direction when calculating \( \Delta v \).
• Keep consistent units throughout calculations.
• Remember acceleration can occur with direction change even if speed is constant.

Speed is a scalar quantity that measures how fast an object is moving, expressed in metres per second (m/s). Acceleration is the rate of change of velocity with time, measured in metres per second squared (m/s²). It can be positive (speed increasing) or negative (speed decreasing, also called deceleration).

Acceleration is calculated using the formula: a = Δv / Δt, where Δv is the change in velocity (m/s) and Δt is the time interval (s).

• Speed (v) units: metres per second (m/s)
• Acceleration (a) units: metres per second squared (m/s²)
• Acceleration formula: a = (final velocity − initial velocity) / time taken

A speed-time graph plots speed (vertical axis, m/s) against time (horizontal axis, seconds). The shape and slope of the graph provide information about the object's motion:

A horizontal line means speed is constant, so acceleration is zero.

A straight line with a positive slope means constant acceleration (speed increasing steadily). The gradient equals the acceleration value (m/s²). The steeper the slope, the greater the acceleration magnitude.

• Vertical axis: Speed (m/s)
• Horizontal axis: Time (s)
• Horizontal line: constant speed, zero acceleration
• Straight line with positive slope: constant positive acceleration

A straight line with a negative slope indicates constant deceleration (speed decreasing steadily). The gradient is negative and equals the acceleration (negative value).

A curved line on a speed-time graph shows changing acceleration. The slope (gradient) is not constant but varies with time, indicating acceleration is increasing or decreasing.

The steeper the curve becomes, the greater the acceleration or deceleration at that moment.

• Straight line with negative slope: constant deceleration
• Curved line: changing acceleration
• Gradient at any point = instantaneous acceleration

To calculate acceleration from a speed-time graph with constant acceleration:

1. Choose two points on the straight line: (t₁, v₁) and (t₂, v₂).

2. Calculate the change in speed: Δv = v₂ − v₁ (m/s). Note the sign: positive for acceleration, negative for deceleration.

• Acceleration a = Δv / Δt = (v₂ − v₁) / (t₂ − t₁)
• Units: m/s² (metres per second squared)
• Example: If speed changes from 5 m/s to 15 m/s in 4 seconds, a = (15 − 5) / 4 = 2.5 m/s²

Given a table of speed and time data: to determine if acceleration is constant:

Calculate acceleration between successive time intervals using a = Δv / Δt.

If acceleration values are the same (or very close), acceleration is constant. If they vary significantly, acceleration is changing.

• Calculate acceleration for each interval
• Compare acceleration values
• Constant acceleration if values are equal within measurement uncertainty
• Changing acceleration if values differ noticeably

A speed-time graph plots speed (vertical axis) against time (horizontal axis). The speed is measured in metres per second (m/s) and time in seconds (s).

The shape and slope of the graph provide information about the motion of an object:

A horizontal line indicates constant speed (zero acceleration). An upward sloping line indicates increasing speed (positive acceleration). A downward sloping line indicates decreasing speed (negative acceleration or deceleration).

• Vertical axis: Speed (m/s)
• Horizontal axis: Time (s)
• Slope (gradient) = change in speed ÷ change in time

Acceleration is defined as the rate of change of speed with time. It is a vector quantity but here we consider only magnitude since speed is scalar.

Mathematically, acceleration (a) is given by: \[ a = \frac{\Delta v}{\Delta t} \] where \( \Delta v \) is the change in speed (final speed - initial speed) in metres per second (m/s), and \( \Delta t \) is the time interval in seconds (s).

On a speed-time graph, acceleration corresponds to the slope (gradient) of the line segment between two points. A steeper slope means greater acceleration.

• Acceleration (a) = change in speed (Δv) ÷ time taken (Δt)
• Units of acceleration: metres per second squared (m/s²)
• Slope of speed-time graph = acceleration

To calculate acceleration from a speed-time graph:

1. Identify two points on the straight line section of the graph. Record their coordinates as (t₁, v₁) and (t₂, v₂).

2. Calculate the change in speed: \( \Delta v = v_2 - v_1 \). 3. Calculate the change in time: \( \Delta t = t_2 - t_1 \). 4. Use the formula: \[ a = \frac{\Delta v}{\Delta t} = \frac{v_2 - v_1}{t_2 - t_1} \] 5. Include units in the final answer (m/s²).

• Select two points on the graph
• Calculate Δv and Δt
• Apply a = Δv / Δt
• Express answer with correct units (m/s²)

A car's speed increases from 10 m/s to 30 m/s over 5 seconds. Calculate its acceleration.

Step 1: Identify values: Initial speed, \( v_1 = 10 \text{ m/s} \) Final speed, \( v_2 = 30 \text{ m/s} \) Time interval, \( \Delta t = 5 \text{ s} \)

Step 2: Calculate change in speed: \( \Delta v = 30 - 10 = 20 \text{ m/s} \) Step 3: Calculate acceleration: \[ a = \frac{\Delta v}{\Delta t} = \frac{20}{5} = 4 \text{ m/s}^2 \] Step 4: Write final answer: Acceleration = 4 m/s² (positive, indicating speeding up)

• Identify initial and final speeds and time interval
• Calculate Δv and Δt
• Substitute into formula a = Δv / Δt
• Answer: 4 m/s²

Forgetting to subtract initial speed from final speed correctly (order matters).

Using the wrong units or omitting units in the final answer.

Calculating the slope using incorrect points or points not on the straight line section (only use linear sections). Confusing speed-time graphs with distance-time graphs (acceleration is slope of speed-time, not distance-time).

• Always subtract initial speed from final speed (v₂ - v₁)
• Check units carefully (m/s for speed, s for time)
• Use points on a straight line section for constant acceleration
• Remember acceleration is slope of speed-time graph

Acceleration is defined as the rate of change of velocity with respect to time. It is a vector quantity, meaning it has both magnitude and direction.

Deceleration refers specifically to acceleration that causes a decrease in the speed of an object. It is therefore acceleration with a negative value when considering the direction of velocity.

In physics, deceleration is not a separate quantity but is negative acceleration. This means the velocity of the object is reducing over time.

• Acceleration (a) = change in velocity (Δv) ÷ time taken (Δt)
• Deceleration means a < 0 (negative acceleration)
• Deceleration causes speed to decrease

The formula for acceleration is: a = (v - u) / t, where:

a is acceleration (m/s²) which can be negative for deceleration,

u is the initial velocity (m/s), v is the final velocity (m/s), and t is the time taken for the change (s).

• If v < u, then a is negative, indicating deceleration.
• Example calculation: A car slows from 20 m/s to 5 m/s in 3 seconds.
• Calculate acceleration: a = (5 - 20) / 3 = -15 / 3 = -5 m/s².
• The negative sign shows the car is decelerating.

Since velocity is a vector, acceleration can be positive or negative depending on the direction of velocity change.

Deceleration occurs when acceleration acts opposite to the direction of motion, reducing speed.

For example, a ball thrown upwards slows down due to acceleration caused by gravity acting downwards, which is deceleration relative to the upward motion.

• Deceleration = acceleration opposite to velocity direction
• Negative acceleration reduces speed
• Positive acceleration increases speed

On a speed-time graph, speed is plotted on the vertical axis (m/s) and time on the horizontal axis (s).

Deceleration is shown by a downward sloping line, indicating speed decreases over time.

The gradient of the line equals acceleration; a negative gradient means deceleration.

• Speed-time graph axes: speed (m/s) vs time (s)
• Negative gradient = deceleration
• Area under graph = distance traveled

A common mistake is to confuse deceleration with acceleration and ignore the negative sign.

Always consider the direction of velocity change to determine if acceleration is positive or negative.

When calculating acceleration, carefully subtract initial velocity from final velocity and include units to avoid sign errors.

• Deceleration is negative acceleration, not zero acceleration
• Check velocity direction and values carefully
• Include units in calculations to avoid mistakes

When an object is dropped in a uniform gravitational field near Earth's surface, it experiences a downward force due to gravity called weight, calculated by W = mg, where m is the mass in kilograms and g is the acceleration due to gravity (approximately 9.8 m/s²). This force causes the object to accelerate downwards.

As the object falls through air or any fluid, it experiences an upward force called air resistance or drag. This force opposes the motion and increases with the object's speed and the properties of the fluid (such as density and viscosity).

• Weight (W) = mg, acts downwards.
• Air resistance acts upwards, opposing motion.
• Air resistance depends on speed, shape, and fluid properties.

At the start of the fall, the object’s speed is zero, so air resistance is negligible. The net force is approximately equal to the weight, causing the object to accelerate downwards at nearly 9.8 m/s².

As the object speeds up, air resistance increases, reducing the net downward force and thus the acceleration.

Eventually, the upward air resistance force grows to match the downward weight, reducing the net force to zero. At this point, acceleration ceases.

• Initial acceleration ≈ g (9.8 m/s²).
• Air resistance increases with speed, reducing acceleration.
• Net force = weight − air resistance.

Terminal velocity is the constant speed reached when the upward air resistance force equals the downward gravitational force, resulting in zero net force and zero acceleration.

At terminal velocity, the object continues to fall at a steady speed without speeding up or slowing down.

Terminal velocity depends on factors such as the object's mass, shape, surface area, and the density of the fluid it falls through.

• Terminal velocity occurs when weight = air resistance.
• At terminal velocity, acceleration = 0 and speed is constant.
• Heavier or more streamlined objects have higher terminal velocities.

The velocity-time graph starts at zero velocity and shows a curve where velocity increases rapidly at first, then the gradient (acceleration) decreases as air resistance grows.

The graph levels off to a horizontal line at terminal velocity, indicating constant speed and zero acceleration.

The gradient of the graph at any point represents the acceleration at that time.

• Vertical axis: velocity (m/s).
• Horizontal axis: time (s).
• Curve rises steeply then flattens at terminal velocity.

Objects with large surface areas relative to their mass, such as a feather or parachute, experience greater air resistance and reach lower terminal velocities.

Dense, compact objects like a metal ball fall faster and reach higher terminal velocities.

Understanding terminal velocity is important in fields such as skydiving, where parachutes increase air resistance to reduce terminal velocity for safe landing.

• Surface area affects air resistance and terminal velocity.
• Mass and shape influence falling speed.
• Applications include parachutes and raindrops.