Kinematics

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Distance & Displacement

Distance is a measure of how far an object travels, and it is a scalar quantity (direction is not important).
Displacement is a measure of how far something is from its starting position, along with its direction; it is a vector quantity.
Displacement describes the change in position and includes both magnitude and direction.
For a journey that returns to the starting point, the total distance is non-zero but the displacement is zero.
When travelling to school, the distance includes all the twists and turns of the roads, while the displacement is the straight-line distance from home to school.

Distance vs displacement

Speed & Velocity

Speed is the distance travelled per second; it is a scalar quantity.
Average speed = total distance ÷ time taken.
Velocity is the rate of change of displacement; it is a vector quantity.
Velocity is speed in a given direction, so it can be positive or negative.
Instantaneous speed (or velocity) is the speed (or velocity) at a particular moment in time.
On a displacement–time graph, the gradient gives velocity; for a curved line, draw a tangent at the required time and calculate its gradient.
Average velocity = total displacement ÷ total time taken.
If acceleration is constant, average velocity can also be calculated as (u + v)/2.

Comparing speed and velocity

Acceleration

Acceleration is defined as the rate of change of velocity.
Acceleration is a vector quantity measured in metres per second squared (m s⁻²).
Average acceleration = change in velocity ÷ time taken, or a = Δv/Δt.
Change in velocity = final velocity − initial velocity, or Δv = v − u.
If an object is speeding up, acceleration is positive; if slowing down, acceleration is negative (deceleration).
Acceleration can also be negative if the object is accelerating in the negative direction.
Instantaneous acceleration is the acceleration at a particular point in time; it is shown by a curved line on a velocity–time graph.

Illustration of positive and negative acceleration with a rocket and a car.

Kinematic Equations

The kinematic equations (SUVAT equations) describe motion with constant (uniform) acceleration.
The five variables are: s = displacement, u = initial velocity, v = final velocity, a = acceleration, t = time interval.
The four equations are: v = u + at; s = ut + ½at²; v² = u² + 2as; s = (u + v)t / 2.
These equations are given in the data booklet, so you do not need to memorise them.
'Starts from rest' means u = 0 and t = 0.
'Falling due to gravity' means a = g = 9.8 m s⁻²; choose a positive direction and keep it consistent.
To solve problems: list known and unknown quantities, choose the equation containing those quantities, convert units to SI, then substitute and solve.

Motion Graphs

On a displacement–time graph: slope equals velocity; a straight diagonal line represents constant velocity; a curved line represents acceleration.
On a displacement–time graph, a positive slope means motion in the positive direction, a negative slope means motion in the negative direction, and a zero slope means the object is at rest.
On a velocity–time graph: slope equals acceleration; a straight diagonal line represents uniform acceleration; a curved line represents non-uniform acceleration.
On a velocity–time graph, the area under the curve equals the change in displacement.
On an acceleration–time graph: the area under the curve equals the change in velocity; a horizontal line represents constant acceleration.
For a bouncing ball, acceleration due to gravity is always directed downwards; at the highest point the velocity is momentarily zero and changes direction.

Motion on a speed–time graph

Projectile Motion

A projectile is a particle moving freely under gravity in a two-dimensional plane, with negligible fluid resistance and constant g.
The horizontal and vertical components of motion are independent of each other and must be analysed separately using the SUVAT equations.
Horizontal motion: velocity is constant, acceleration is zero.
Vertical motion: acceleration is g (9.8 m s⁻²), acting downwards.
If a projectile is launched with speed u at angle θ to the horizontal, its initial horizontal component is u cosθ and its initial vertical component is u sinθ.
Time of flight is the total time in the air; for level ground, the time to maximum height is half the total time.
Maximum height is reached when the vertical velocity component is zero.
Range is the horizontal distance travelled by the projectile.

Fluid Resistance

Fluid resistance refers to the resistive forces (viscous drag) acting on an object moving through a gas or liquid.
Viscous drag is a type of friction that always acts in the opposite direction to motion and increases with the object's speed.
Lift is an upward force perpendicular to the fluid flow, often arising from Newton's Third Law.
In projectile motion, air resistance decreases the horizontal velocity, range, maximum height and time of flight.
With air resistance, the trajectory is no longer a parabola; it is steeper on the way down than on the way up.
For sports like long jump or javelin, an optimum angle is used to maximise range against air resistance.

Air resistance is a type of drag: it opposes motion through a fluid and heats the object.

Terminal Speed

For a body falling in a vacuum, the only force is weight, so acceleration is g.
As a body falls through a fluid, viscous drag increases with speed, reducing the resultant force and therefore the acceleration.
Terminal velocity is reached when the viscous drag force equals the weight, so the resultant force is zero and the body falls at a constant velocity.
On a velocity–time graph for a skydiver, the gradient (acceleration) decreases until it becomes zero at terminal velocity.
A heavier skydiver reaches a higher terminal velocity and reaches it faster than a lighter skydiver with the same surface area.
After a parachute is deployed, the skydiver decelerates to a lower terminal velocity; they do not move upwards.

A speed-time graph for a skydiver: air resistance builds as speed increases until it balances weight and a constant (terminal) velocity is reached.

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練習問題

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1.What is the definition of acceleration?
Easy
- AThe rate of change of velocity
- BThe rate of change of displacement
- CThe length between two points
- DThe length between two points in a certain direction
2.Which of the following correctly describes distance and displacement?
Easy
- ADistance is a scalar and displacement is a vector
- BDistance is a vector and displacement is a scalar
- CBoth distance and displacement are scalars
- DBoth distance and displacement are vectors
3.A sprint walker completes a 200 m race in 40 s. What is their average speed during the race?
Easy
- A5 m s⁻¹
- B6 m s⁻¹
- C7 m s⁻¹
- D8 m s⁻¹
4.Velocity is a vector quantity.
Easy
True or false?
5.What does the gradient of a displacement-time graph represent?
Easy
- AVelocity
- BDistance
- CSpeed
- DAcceleration
6.A velocity-time graph is shown for an object. Which property of the graph represents the total displacement of the object?
Easy
- AThe total area between the line and the axis
- BThe gradient of the line
- CThe y-intercept
- DThe x-intercept
7.A car accelerates from rest to a speed of 40 m s⁻¹ in 5 seconds. What is the car's acceleration?
Medium
- A8 m s⁻²
- B5 m s⁻²
- C20 m s⁻²
- D35 m s⁻²
8.Which of the following quantities are vectors? (Select all that apply.)
Medium
- ADisplacement
- BVelocity
- CAcceleration
- DDistance
- ESpeed

もっと問題

9.An object moves through one complete circle from point W to X to Y to Z and finally back to W. The distance between X and Z is 6 km. Which row is correct about the displacement and distance of the object?
Medium
- ADisplacement = 0 km, Distance = 6π km
- BDisplacement = 0 km, Distance = 9π km
- CDisplacement = 6π km, Distance = 0 km
- DDisplacement = 6π km, Distance = 9π km
10.A ball starts from rest and moves horizontally. Five positions of the ball are shown at time intervals of 0.50 ms. The horizontal distance between X, the initial position, and Y, the final position, is 0.060 m. What is the average acceleration of the ball between X and Y?
Medium
- A48 m s⁻²
- B60 m s⁻²
- C19200 m s⁻²
- D30000 m s⁻²
11.An object is released from rest near the surface of the Earth and allowed to fall freely. Which of the following correctly describes the speed and acceleration before the object reaches terminal velocity?
Medium
- ASpeed increases, acceleration decreases
- BSpeed remains constant, acceleration remains constant
- CSpeed remains constant, acceleration decreases
- DSpeed increases, acceleration remains constant
12.An object is released from a stationary hot air balloon at height s above the ground. Air resistance is negligible. An identical object is released at the same height above the ground from another balloon that is rising at a constant speed. Which of the following does not increase for the object released from the rising balloon?
Hard
- AIts acceleration
- BThe time taken for it to reach the ground
- CThe distance through which it falls
- DThe speed with which it hits the ground
13.A projectile is fired from level ground with speed v at an angle θ to the ground. Ignoring air resistance, which of the following is a correct expression for the maximum height reached by the projectile?
Hard
- Av² sin²θ / (2g)
- Bv² sin²θ / g
- Cv sin²θ / (2g)
- Dv sin²θ / g
14.A small uniform bob is attached to a simple pendulum and set into motion from some angular displacement. When the bob passes through the lowest point of its motion, the pendulum is cut. If at this instant the bob is moving with a speed v and is at a height h above ground, how far away does the bob land from a position measured directly under the point at which the pendulum is cut?
Hard
- Av √(2h/g)
- Bv √(h/g)
- Cv h / g
- Dv √(h/(2g))
15.Which of the following quantities are scalar? (Select all that apply.)
Medium
- ADistance
- BDisplacement
- CSpeed
- DVelocity
- EAcceleration
16.A velocity-time graph shows a straight line with a positive slope. Which statements about the motion are correct? (Select all that apply.)
Medium
- AThe acceleration is constant.
- BThe velocity is increasing.
- CThe object is stationary.
- DThe displacement is given by the area under the line.
- EThe acceleration is zero.
17.Which of the following are correct about an object in projectile motion, ignoring air resistance? (Select all that apply.)
Medium
- AThe horizontal velocity is constant.
- BThe vertical acceleration is constant.
- CThe horizontal acceleration is equal to g.
- DAt maximum height, the vertical velocity is zero.
- EThe time to reach maximum height is half the total time of flight.
18.A skydiver jumps from a plane and eventually reaches terminal velocity. Which statements are correct? (Select all that apply.)
Hard
- AAt terminal velocity, the acceleration is zero.
- BThe drag force increases as speed increases.
- CThe terminal velocity is higher for a heavier skydiver with the same surface area.
- DThe skydiver continues to accelerate at g after reaching terminal velocity.
- EA heavier skydiver reaches terminal velocity faster than a lighter one.
19.Which of the following are correct units for the quantities listed? (Select all that apply.)
Easy
- AAcceleration: m s⁻²
- BVelocity: m s⁻¹
- CDisplacement: m
- DSpeed: m s⁻²
- ETime: s
20.Match each quantity with its correct definition.
Medium
- Distance
- Displacement
- Speed
- Velocity
- Acceleration
- The total length of the path travelled
- The change in position, including direction
- The rate of change of distance
- The rate of change of displacement
- The rate of change of velocity
21.Match each graph feature with the quantity it represents.
Medium
- Gradient of a displacement-time graph
- Area under a velocity-time graph
- Gradient of a velocity-time graph
- Area under an acceleration-time graph
- Gradient of an acceleration-time graph
- Velocity
- Change in displacement
- Acceleration
- Change in velocity
- Meaningless
22.Match each term with its correct description in the context of projectile motion.
Medium
- Time of flight
- Maximum height
- Range
- Horizontal velocity
- Vertical velocity at maximum height
- Total time the projectile is in the air
- The highest point reached, where vertical velocity is zero
- The horizontal distance travelled
- Remains constant throughout the motion (ignoring air resistance)
- Zero
23.A ball is thrown vertically upwards and then falls back down. Ignoring air resistance, arrange the stages of its motion in chronological order.
Medium
- Ball leaves hand with upward velocity
- Ball reaches maximum height, velocity is zero
- Ball accelerates downwards due to gravity
- Ball falls back to the starting height
24.Arrange the stages of a skydiver's fall in the correct order.
Hard
- Jumps from plane and accelerates due to gravity
- Drag force increases as speed increases
- Drag force equals weight, terminal velocity reached
- Parachute opens, decelerates to a lower terminal velocity
25.Distance is a vector quantity.
Easy
True or false?
26.The area under a velocity-time graph represents the change in displacement.
Easy
True or false?
27.In projectile motion, the horizontal component of velocity changes due to gravity.
Medium
True or false?
28.When a skydiver reaches terminal velocity, the acceleration is zero.
Medium
True or false?
29.For a projectile launched at an angle to the horizontal, the time to reach maximum height is half the total time of flight.
Medium
True or false?
30.A car is travelling along a straight road. Its velocity increases from 10 m s⁻¹ to 24 m s⁻¹ in 7 seconds. What is the magnitude of the car's average acceleration?
Medium
- A2 m s⁻²
- B4.9 m s⁻²
- C14 m s⁻²
- D34 m s⁻²
31.A stone is dropped from rest from the top of a cliff and falls freely under gravity (g = 9.8 m s⁻²). After 3.0 s, what is the stone's speed?
Medium
- A9.8 m s⁻¹
- B14.7 m s⁻¹
- C29.4 m s⁻¹
- D44.1 m s⁻¹
32.A ball is thrown vertically upwards with an initial speed of 12 m s⁻¹. Ignoring air resistance and taking g = 9.8 m s⁻², what is the maximum height reached by the ball?
Hard
- A0.61 m
- B1.22 m
- C7.35 m
- D14.7 m
33.The velocity-time graph of an object is a straight line sloping downwards from a positive value to zero. Which row correctly describes the object's acceleration and motion?
Medium
- AAcceleration is positive and the object is speeding up.
- BAcceleration is negative and the object is slowing down.
- CAcceleration is zero and the object is moving at constant velocity.
- DAcceleration is negative and the object is speeding up.
34.A ball is thrown horizontally from the top of a tower and lands on the ground. Air resistance is negligible. Which statement about the ball's motion is correct?
Hard
- AThe horizontal velocity changes because of gravity.
- BThe vertical velocity is constant throughout the flight.
- CThe horizontal and vertical motions are independent of each other.
- DThe ball reaches the ground at the same time as a ball dropped from rest from the same height only if it is thrown slowly.
35.A projectile is launched at an angle to the horizontal. Ignoring air resistance, which of the following is a correct expression for the horizontal range R of the projectile? (u = initial speed, θ = launch angle, g = acceleration due to gravity)
Hard
- AR = u² sin(2θ) / g
- BR = u² sin²θ / (2g)
- CR = u² cos²θ / g
- DR = u sin(2θ) / g
36.A skydiver jumps from a plane and falls towards the ground. Which statement correctly describes the motion when the skydiver reaches terminal velocity?
Medium
- AThe skydiver's weight becomes zero.
- BThe viscous drag force equals the skydiver's weight and the acceleration is zero.
- CThe viscous drag force is greater than the skydiver's weight and the skydiver decelerates.
- DThe skydiver's acceleration is equal to g.
37.A student measures the time for a ball to fall a certain distance and calculates the average speed as distance ÷ time. Which quantity does this calculation give?
Medium
- AInstantaneous speed at the end of the fall
- BAverage speed over the fall
- CAcceleration of the ball
- DDisplacement of the ball
38.Which of the following statements about a projectile in motion with negligible air resistance are correct? (Select all that apply.)
Hard
- AThe horizontal component of velocity remains constant.
- BThe vertical component of velocity changes at a constant rate.
- CThe acceleration is zero at the highest point.
- DThe time to reach maximum height is half the total time of flight.
- EThe horizontal acceleration is equal to g.
39.A ball is thrown vertically upwards and then falls back to the ground. Ignoring air resistance, which of the following are correct at the highest point of its motion? (Select all that apply.)
Hard
- AThe velocity is zero.
- BThe acceleration is zero.
- CThe acceleration is g downwards.
- DThe ball is momentarily at rest.
- EThe speed is maximum.
40.The area under a velocity-time graph represents the change in displacement of the object.
Easy
True or false?
41.A luxurious speed boat travels with an initial velocity of 10 m s⁻¹. It accelerates at 2 m s⁻² for 5 seconds. Calculate the final velocity of the speed boat.
Medium
42.Complete the definitions.
Easy
Distance is a ____ quantity, while displacement is a ____ quantity.
43.A car accelerates uniformly from 10 m s⁻¹ to 30 m s⁻¹ in 5 s. Calculate its acceleration in m s⁻².
Medium
44.A ball is thrown vertically upwards with an initial velocity of 20 m s⁻¹. Calculate the maximum height it reaches in metres. Ignore air resistance and take g = 9.8 m s⁻².
Medium
45.A stone is dropped from rest from a height of 45 m. Calculate the time it takes to reach the ground in seconds. Take g = 9.8 m s⁻².
Hard
46.Complete the sentence about speed and velocity.
Easy
Speed is a ____ quantity, while velocity is a ____ quantity.
47.Complete the sentence about acceleration.
Medium
Acceleration is defined as the rate of change of ____.
48.Complete the sentence about projectile motion.
Medium
In projectile motion, the horizontal velocity is ____ and the vertical acceleration is ____.
49.Complete the sentence about terminal velocity.
Medium
Terminal velocity is reached when the ____ force equals the ____ of the object.
50.What is the SI unit of acceleration?
Medium
51.What is the name of the velocity reached when the drag force equals the weight of a falling object?
Medium
52.A cyclist accelerates uniformly from rest at 1.5 m s⁻² for 8 seconds. Calculate the distance travelled in metres.
Medium
53.A ball is thrown upward with an initial velocity of +3 m s⁻¹. Which SUVAT equation will correctly calculate the maximum height reached by the ball? Explain why.
Medium
54.A student designs an experiment to determine the acceleration due to free fall, g, using a falling ball bearing and a pair of light gates. The value obtained is inaccurate. Describe one factor that could have contributed to the inaccurate value of g and explain how it affects the result.
Hard
55.Explain how the gradient of a velocity-time graph can be used to determine the acceleration of an object.
Hard
56.Describe the motion of a skydiver from the moment they jump from a plane until they reach terminal velocity, including the forces involved.
Hard
57.Explain how the motion of a skydiver changes from the moment they jump from a plane until they reach terminal velocity, including the forces acting.
Hard
58.A luxurious speed boat travels with an initial velocity of 10 m s⁻¹. It accelerates at 2 m s⁻² for 5 seconds. Calculate the final velocity of the speed boat.
Medium
- A18 m/s
- B24 m/s
- C22 m/s
- D20 m/s
59.A car accelerates uniformly from 10 m s⁻¹ to 30 m s⁻¹ in 5 s. Calculate its acceleration in m s⁻².
Medium
- A6 m/s²
- B3 m/s²
- C4 m/s²
- D5 m/s²
60.A ball is thrown vertically upwards with an initial velocity of 20 m s⁻¹. Calculate the maximum height it reaches in metres. Ignore air resistance and take g = 9.8 m s⁻².
Medium
- A18.4 m
- B20.4 m
- C24.4 m
- D22.4 m
61.A stone is dropped from rest from a height of 45 m. Calculate the time it takes to reach the ground in seconds. Take g = 9.8 m s⁻².
Hard
- A2.03 s
- B4.03 s
- C3.03 s
- D5.03 s
62.A cyclist accelerates uniformly from rest at 1.5 m s⁻² for 8 seconds. Calculate the distance travelled in metres.
Medium
- A48 m
- B43 m
- C58 m
- D53 m

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