Constant acceleration formulae

A car moves in a straight line along a horizontal road with constant acceleration 2 m s^–2 The car is moving with speed 15 m s^–1 in the direction of the acceleration when it passes a signpost on the road. The car is modelled as a particle.

Use the model to find the speed of the car 4 s after passing the signpost. Figure 1 below shows the horizontal forces acting on the car. Given that

• the car has mass 800 kg

• the driving force of the engine has magnitude D newtons

• the resistance to the motion of the car has magnitude 400 N

• the acceleration of the car is 2 m s^–2 in the direction of the driving force

use the model to find the value of D.

[In this question, i and j are horizontal unit vectors due east and due north respectively.] A particle P of mass 0.5 kg moves with constant acceleration (2i – 2.4j) m s^–2 on a smooth horizontal plane under the action of a constant horizontal force F N.

Find F in terms of i and j. At time t = 0, P is moving with velocity (–7i + 7.8j) m s^–1

Find the velocity of P at time t = 2 seconds.

Find the direction of motion of P at time t = 2 seconds, giving your answer as a bearing in degrees. At time t = 0, P passes through the point O. At time t = 5 seconds, P passes through the point A.

Find OA → in terms of i and j.

Figure 2 shows a speed‑time graph for a model of the motion of an athlete running a 200 m race in 24 s. The athlete

• starts from rest at time t = 0 and accelerates at a constant rate, reaching a speed of

• then moves at a constant speed of 10 m s^–1 fromt = 4 tot = 18

• then decelerates at a constant rate from t = 18 to t = 24, crossing the finishing line

Using the model,

find the acceleration of the athlete during the first 4 s of the race, stating the units of your answer,

find the distance covered by the athlete during the first 18 s of the race,

find the value of U.

A car is initially at rest on a straight horizontal road. The car then accelerates along the road with a constant acceleration of 3.2 m s^–2 Find

the speed of the car after 5 s,

the distance travelled by the car in the first 5 s.

[In this question, i and j are horizontal unit vectors and position vectors are given relative to a fixed origin O ] A particle P is moving on a smooth horizontal plane. The particle has constant acceleration (2.4i + j) m s^–2 At time t = 0, P passes through the point A. At time t = 5 s, P passes through the point B. The velocity of P as it passes through A is (–16i – 3j) m s^–1

Find the speed of P as it passes through B. The position vector of A is (44i – 10j) m. At time t = T seconds, where T 5, P passes through the point C. The position vector of C is (4i + cj) m.

Find the value of T.

Find the value of c.

A particle P moves with constant acceleration (2i − 3j) m s⁻²

At time t = 0, P is moving with velocity 4im s⁻¹

Find the velocity of P at time t = 2 seconds.

At time t = 0, the position vector of P relative to a fixed origin O is (i + j) m.

Find the position vector of P relative to O at time t = 3 seconds.

A particle P moves with constant acceleration (2i − 3j) m s⁻² At time t = 0, P is moving with velocity 4im s⁻¹

Find the velocity of P at time t = 2 seconds. At time t = 0, the position vector of P relative to a fixed origin O is (i + j) m.

Find the position vector of P relative to O at time t = 3 seconds.

A small stone is projected with speed 65 m s⁻¹ from a point O at the top of a vertical cliff.

Point O is 70 m vertically above the point N.

Point N is on horizontal ground. The stone is projected at an angle α above the horizontal, where tanα =

The stone hits the ground at the point A, as shown in Figure 3.

The stone is modelled as a particle moving freely under gravity.

The acceleration due to gravity is modelled as having magnitude 10 m s⁻²

Using the model,

find the time taken for the stone to travel from O to A,

find the speed of the stone at the instant just before it hits the ground at A.

One limitation of the model is that it ignores air resistance.

State one other limitation of the model that could affect the reliability of your answers.

A particle P moves with acceleration (4i − 5j) m s⁻² At time t = 0, P is moving with velocity (−2i + 2j) m s⁻¹

Find the velocity of P at time t = 2 seconds. At time t = 0, P passes through the origin O. At time t = T seconds, where T 0, the particle P passes through the point A. The position vector of A is (λi − 4.5j)m relative to O, where λ is a constant.

Find the value of T.

Hence find the value of λ

A ball of mass 0.2 kg is projected vertically downwards with speed U m s^–1 from a point A which is 2.5 m above horizontal ground. The ball hits the ground. Immediately after hitting the ground, the ball rebounds vertically with a speed of 10 m s^–1. The ball receives an impulse of magnitude 7 N s in its impact with the ground. By modelling the ball as a particle and ignoring air resistance, find

the value of U.

After hitting the ground, the ball moves vertically upwards and passes through a point B which is 1 m above the ground.

Find the time between the instant when the ball hits the ground and the instant when the ball first passes through B.

Sketch a velocity-time graph for the motion of the ball from when it was projected from A to when it first passes through B. (You need not make any further calculations to draw this sketch.)

A particle P of mass 4 m is held at rest at the point X on the surface of a rough inclined plane which is fixed to horizontal ground. The point X is a distance h from the bottom of the inclined plane. The plane is inclined to the horizontal at an angle α where tan . The coefficient of friction between P and the plane is The particle P is attached to one end of a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of the plane. The other end of the string is attached to a particle Q of mass m which hangs freely at a distance d, where d h, below the pulley, as shown in Figure 4. The string lies in a vertical plane through a line of greatest slope of the inclined plane. The system is released from rest with the string taut and P moves down the plane. For the motion of the particles before P hits the ground,

state which of the information given above implies that the magnitudes of the accelerations of the two particles are the same,

write down an equation of motion for each particle,

find the acceleration of each particle.

When P hits the ground, it immediately comes to rest. Given that Q comes to instantaneous rest before reaching the pulley,

A cyclist is moving along a straight horizontal road and passes a point A. Five seconds later, at the instant when she is moving with speed 10 ms^–1, she passes the point She moves with constant acceleration from A to B. Given that AB = 40m, find

the acceleration of the cyclist as she moves from A to B,

the time it takes her to travel from A to the midpoint of

Two particles, A and B, have masses 2 m and m respectively. The particles are attached to the ends of a light inextensible string. Particle A is held at rest on a fixed rough horizontal table at a distance d from a small smooth light pulley which is fixed at the edge of the table at the point P. The coefficient of friction between A and the table is , where < The string is parallel to the table from A to P and passes over the pulley. Particle B hangs freely at rest vertically below P with the string taut and at a height h, (h < d), above a horizontal floor, as shown in Figure 3. Particle A is released from rest with the string taut and slides along the table.

Write down an equation of motion for

Write down an equation of motion for

Hence show that, until B hits the floor, the acceleration of A is 1 2μ).

Find, in terms of g, h and μ, the speed of A at the instant when B hits the floor.

After B hits the floor, A continues to slide along the table. Given that — and that A comes to rest at P,

find d in terms of h.

Describe what would happen if

Two trains M and N are moving in the same direction along parallel straight horizontal tracks. At time t = 0, M overtakes N whilst they are travelling with speeds 40 m s^–1 and 30 m s^–1 respectively. Train M overtakes train N as they pass a point X at the side of the tracks . After overtaking N, train M maintains its speed of 40 m s^–1 for T seconds and then decelerates uniformly, coming to rest next to a point Y at the side of the tracks. After being overtaken, train N maintains its speed of 30 m s^–1 for 25 s and then decelerates uniformly, also coming to rest next to the point Y. The times taken by the trains to travel between X and Y are the same.

Sketch, on the same diagram, the speed-time graphs for the motions of the two trains between X and Y.

Given that XY = 975 m,

find the value of T.

Two forces F₁ and F₂ act on a particle P. The force F₁ is given by F₁ = (–i + 2j) N and F₂ acts in the direction of the vector (i + j). Given that the resultant of F₁ and F₂ acts in the direction of the vector (i + 3j),

find F₂

The acceleration of P is (3i + 9j) m s^–2. At time t = 0, the velocity of P is (3i – 22j) m s^–1

Find the speed of P when t = 3 seconds.

A car starts from rest and moves with constant acceleration along a straight horizontal road. The car reaches a speed of V m s^–1 in 20 seconds. It moves at constant speed V m s^–1 for the next 30 seconds, then moves with constant deceleration m s^–2 until it has speed 8 m s^–1. It moves at speed 8 m s^–1 for the next 15 seconds and then moves with constant deceleration m s^–2 until it comes to rest.

Sketch, in the space below, a speed-time graph for this journey.

In the first 20 seconds of this journey the car travels 140 m. Find

the value of V,

the total time for this journey,

the total distance travelled by the car.

At time t = 0, a particle is projected vertically upwards with speed u from a point A. The particle moves freely under gravity. At time T the particle is at its maximum height H above A.

Find T in terms of u and g.

The point A is at a height 3 H above the ground.

Find, in terms of T, the total time from the instant of projection to the instant when the particle hits the ground.

Figure 2

Two particles A and B have masses 2 m and 3 m respectively. The particles are connected by a light inextensible string which passes over a smooth light fixed pulley. The system is held at rest with the string taut. The hanging parts of the string are vertical and A and B are above a horizontal plane, as shown in Figure 2. The system is released from rest.

Show that the tension in the string immediately after the particles are released

After descending 1.5 m, B strikes the plane and is immediately brought to rest. In the subsequent motion, A does not reach the pulley.

Find the distance travelled by A between the instant when B strikes the plane and the instant when the string next becomes taut.

Given that m = 0.5 kg,

find the magnitude of the impulse on B due to the impact with the plane.

At time t = 0, two balls A and B are projected vertically upwards. The ball A is projected vertically upwards with speed 2 m s^–1 from a point 50 m above the horizontal ground. The ball B is projected vertically upwards from the ground with speed 20 m s^–1. At time t = T seconds, the two balls are at the same vertical height, h metres, above the ground. The balls are modelled as particles moving freely under gravity. Find

the value of T,

the value of h.

A lorry is moving along a straight horizontal road with constant acceleration. The lorry passes a point A with speed u m s^–1, (u < 34), and 10 seconds later passes a point B with speed 34 m s^–1. Given that AB = 240 m, find

the value of u,

the time taken for the lorry to move from A to the mid-point of AB.

A particle P of mass 0.6 kg slides with constant acceleration down a line of greatest slope of a rough plane, which is inclined at 25° to the horizontal. The particle passes through two points A and B, where AB = 10 m, as shown in Figure 3. The speed of P at A is 2 m s^–1. The particle P takes 3.5 s to move from A to B. Find

the speed of P at B,

the acceleration of P,

the coefficient of friction between P and the plane.

A car is travelling along a straight horizontal road. The car takes 120 s to travel between two sets of traffic lights which are 2145 m apart. The car starts from rest at the first set of traffic lights and moves with constant acceleration for 30 s until its speed is 22 m s^–1. The car maintains this speed for T seconds. The car then moves with constant deceleration, coming to rest at the second set of traffic lights.

Sketch, in the space below, a speed-time graph for the motion of the car between the two sets of traffic lights.

Find the value of T.

A motorcycle leaves the first set of traffic lights 10 s after the car has left the first set of traffic lights. The motorcycle moves from rest with constant acceleration, a m s^–2, and passes the car at the point A which is 990 m from the first set of traffic lights. When the motorcycle passes the car, the car is moving with speed 22 m s^–1.

Find the time it takes for the motorcycle to move from the first set of traffic lights to the point A.

Find the value of a.

A truck of mass 1750 kg is towing a car of mass 750 kg along a straight horizontal road. The two vehicles are joined by a light towbar which is inclined at an angle to the road, as shown in Figure 4. The vehicles are travelling at 20 m s^–1 as they enter a zone where the speed limit is 14 m s^–1. The truck’s brakes are applied to give a constant braking force on the truck. The distance travelled between the instant when the brakes are applied and the instant when the speed of each vehicle is 14 m s^–1 is 100 m.

Find the deceleration of the truck and the car.

The constant braking force on the truck has magnitude R newtons. The truck and the car also experience constant resistances to motion of 500 N and 300 N respectively. Given that cos = 0.9, find

the force in the towbar,

the value of R.

The velocity-time graph in Figure 4 represents the journey of a train P travelling along a straight horizontal track between two stations which are 1.5 km apart. The train P leaves the first station, accelerating uniformly from rest for 300 m until it reaches a speed of 30 m s^–1. The train then maintains this speed for T seconds before decelerating uniformly at 1.25 m s^–2, coming to rest at the next station.

Find the acceleration of P during the first 300 m of its journey.

Find the value of T.

A second train Q completes the same journey in the same total time. The train leaves the first station, accelerating uniformly from rest until it reaches a speed of V m s^–1 and then immediately decelerates uniformly until it comes to rest at the next station.

Sketch on the diagram above, a velocity-time graph which represents the journey of train Q.

Find the value of V.

Figure 5 shows two particles A and B, of mass 2 m and 4 m respectively, connected by a light inextensible string. Initially A is held at rest on a rough inclined plane which is fixed to horizontal ground. The plane is inclined to the horizontal at an angle , where tan . The coefficient of friction between A and the plane is . The string passes over a small smooth pulley P which is fixed at the top of the plane. The part of the string from A to P is parallel to a line of greatest slope of the plane and B hangs vertically below P. The system is released from rest with the string taut, with A at the point X and with B at a height h above the ground. For the motion until B hits the ground,

give a reason why the magnitudes of the accelerations of the two particles are the same,

write down an equation of motion for each particle,

find the acceleration of each particle.

Particle B does not rebound when it hits the ground and A continues moving up the plane towards P. Given that A comes to rest at the point Y, without reaching P,

find the distance XY in terms of h.

A car is moving on a straight horizontal road. At time t = 0, the car is moving with speed 20 m s^–1 and is at the point A. The car maintains the speed of 20 m s^–1 for 25 s. The car

8 m s^–1. The car then moves with constant speed 8 m s^–1 for 60 s. The car then moves with constant acceleration until it is moving with speed 20 m s^–1 at the point B.

Sketch a speed-time graph to represent the motion of the car from A to B.

Find the time for which the car is decelerating.

Given that the distance from A to B is 1960 m,

find the time taken for the car to move from A to B.

A particle P is projected vertically upwards from a point A with speed u m s^–1. The point A is 17.5 m above horizontal ground. The particle P moves freely under gravity until it reaches the ground with speed 28 m s^–1.

Show that u = 21

At time t seconds after projection, P is 19 m above A.

Find the possible values of t.

The ground is soft and, after P reaches the ground, P sinks vertically downwards into the ground before coming to rest. The mass of P is 4 kg and the ground is assumed to exert a constant resistive force of magnitude 5000 N on P.

Find the vertical distance that P sinks into the ground before coming to rest.

Two particles P and Q, of mass 0.3 kg and 0.5 kg respectively, are joined by a light horizontal rod. The system of the particles and the rod is at rest on a horizontal plane. At time t = 0, a constant force F of magnitude 4 N is applied to Q in the direction PQ, as shown in Figure 3. The system moves under the action of this force until t = 6 s. During the motion, the resistance to the motion of P has constant magnitude 1 N and the resistance to the motion of Q has constant magnitude 2 N. Find

the acceleration of the particles as the system moves under the action of F,

the speed of the particles at t = 6 s,

the tension in the rod as the system moves under the action of F.

At t = 6 s, F is removed and the system decelerates to rest. The resistances to motion are unchanged. Find

the distance moved by P as the system decelerates,

the thrust in the rod as the system decelerates.

At time t = 0 a ball is projected vertically upwards from a point O and rises to a maximum height of 40 m above O. The ball is modelled as a particle moving freely under gravity.

Show that the speed of projection is 28 m s^–1.

Find the times, in seconds, when the ball is 33.6 m above O.

A girl runs a 400 m race in a time of 84 s. In a model of this race, it is assumed that, starting from rest, she moves with constant acceleration for 4 s, reaching a speed of 5 m s^–1. She maintains this speed for 60 s and then moves with constant deceleration for 20 s, crossing the finishing line with a speed of V m s^–1.

Sketch, in the space below, a speed-time graph for the motion of the girl during the whole race.

Find the distance run by the girl in the first 64 s of the race.

Find the value of V.

Find the deceleration of the girl in the final 20 s of her race.

Two particles P and Q have masses 0.3 kg and m kg respectively. The particles are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of a fixed rough plane. The plane is inclined to the horizontal at

The string lies in a vertical plane through a line of greatest slope of the inclined plane. The particle P is held at rest on the inclined plane and the particle Q hangs freely below the pulley with the string taut, as shown in Figure 2.

Find

the magnitude of the normal reaction of the inclined plane on P,

the value of m.

When the particles have been moving for 0.5 s, the string breaks. Assuming that P does not reach the pulley,

find the further time that elapses until P comes to instantaneous rest.

A ball is thrown vertically upwards with speed u m s¹ from a point P at height h metres above the ground. The ball hits the ground 0.75 s later. The speed of the ball immediately before it hits the ground is 6.45 m s¹. The ball is modelled as a particle.

Find the height above P to which the ball rises before it starts to fall towards the ground again.

Find the value of h.

A car accelerates uniformly from rest for 20 seconds. It moves at constant speed v m ss¹ for the next 40 seconds and then decelerates uniformly for 10 seconds until it comes to rest.

For the motion of the car, sketch

a speed-time graph,

an acceleration-time graph.

Given that the total distance moved by the car is 880 m,

find the value of v.

Two particles A and B, of mass 7 kg and 3 kg respectively, are attached to the ends of a light inextensible string. Initially B is held at rest on a rough fixed plane inclined at angle to the horizontal, where tan = . The part of the string from B to P is parallel to a line of greatest slope of the plane. The string passes over a small smooth pulley, P, fixed at the top of the plane. The particle A hangs freely below P, as shown in Figure 4. The coefficient of friction between B and the plane is . The particles are released from rest with the string taut and B moves up the plane.

Find the magnitude of the acceleration of B immediately after release.

Find the speed of B when it has moved 1 m up the plane.

When B has moved 1 m up the plane the string breaks. Given that in the subsequent motion B does not reach P,

find the time between the instants when the string breaks and when B comes to instantaneous rest.

Two cars P and Q are moving in the same direction along the same straight horizontal road. Car P is moving with constant speed 25 m s¹. At time t 0, P overtakes Q which is moving with constant speed 20 m s¹. From t T seconds, P decelerates uniformly, coming to rest at a point X which is 800 m from the point where P overtook Q. From t 25 s, Q decelerates uniformly, coming to rest at the same point X at the same instant as P.

Sketch, on the same axes, the speed-time graphs of the two cars for the period from t 0 to the time when they both come to rest at the point X.

Find the value of T.

A ball is projected vertically upwards with a speed of 14.7 m s¹ from a point which is 49 m above horizontal ground. Modelling the ball as a particle moving freely under gravity, find

the greatest height, above the ground, reached by the ball,

the speed with which the ball first strikes the ground,

the total time from when the ball is projected to when it first strikes the ground.

Two particles A and B have mass 0.4 kg and 0.3 kg respectively. The particles are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed above a horizontal floor. Both particles are held, with the string taut, at a height of 1m above the floor, as shown in Figure 3. The particles are released from rest and in the subsequent motion B does not reach the pulley.

Find the tension in the string immediately after the particles are released.

Find the acceleration of A immediately after the particles are released.

When the particles have been moving for 0.5 s, the string breaks.

Find the further time that elapses until B hits the floor.

An athlete runs along a straight road. She starts from rest and moves with constant acceleration for 5 seconds, reaching a speed of 8 m s^–1. This speed is then maintained for T seconds. She then decelerates at a constant rate until she stops. She has run a total of 500 m in 75 s.

In the space below, sketch a speed-time graph to illustrate the motion of the athlete.

Calculate the value of T.

A particle of mass 0.8 kg is held at rest on a rough plane. The plane is inclined at 30° to the horizontal. The particle is released from rest and slides down a line of greatest slope of the plane. The particle moves 2.7 m during the first 3 seconds of its motion. Find

the acceleration of the particle,

the coefficient of friction between the particle and the plane.

The particle is now held on the same rough plane by a horizontal force of magnitude X newtons, acting in a plane containing a line of greatest slope of the plane, as shown in Figure 3. The particle is in equilibrium and on the point of moving up the plane.

Find the value of X.

Three posts P, Q and R, are fixed in that order at the side of a straight horizontal road. The distance from P to Q is 45 m and the distance from Q to R is 120 m. A car is moving along the road with constant acceleration a m s^–2. The speed of the car, as it passes P, is u m s^–1. The car passes Q two seconds after passing P, and the car passes R four seconds after passing Q. Find

the value of u,

the value of a.

At time t = 0, a particle is projected vertically upwards with speed u m s^–1 from a point 10 m above the ground. At time T seconds, the particle hits the ground with speed 17.5 m s^–1. Find

the value of u,

the value of T.

A particle P of mass 0.4 kg moves under the action of a single constant force F newtons. The acceleration of P is (6i + 8j) m s^–2. Find

the angle between the acceleration and i,

the magnitude of F.

At time t seconds the velocity of P is v m s^–1. Given that when t = 0, v = 9i – 10j,

find the velocity of P when t = 5.

A car is moving along a straight horizontal road. The speed of the car as it passes the point A is 25 m s^–1 and the car maintains this speed for 30 s. The car then decelerates uniformly to a speed of 10 m s^–1. The speed of 10 m s^–1 is then maintained until the car passes the point B. The time taken to travel from A to B is 90 s and AB = 1410 m.

Sketch, in the space below, a speed-time graph to show the motion of the car from A to B.

Calculate the deceleration of the car as it decelerates from 25 m s^–1 to 10 m s^–1.

Two particles P and Q, of mass 2 kg and 3 kg respectively, are joined by a light inextensible string. Initially the particles are at rest on a rough horizontal plane with the string taut. A constant force F of magnitude 30 N is applied to Q in the direction PQ, as shown in Figure 4. The force is applied for 3 s and during this time Q travels a distance of 6 m. The coefficient of friction between each particle and the plane is . Find

the acceleration of Q,

the tension in the string.

State how in your calculation you have used the information that the string is inextensible.

When the particles have moved for 3 s, the force F is removed.

Find the time between the instant that the force is removed and the instant that Q comes to rest.

A firework rocket starts from rest at ground level and moves vertically. In the first 3 s of its motion, the rocket rises 27 m. The rocket is modelled as a particle moving with constant acceleration a m s^–2. Find

the value of a,

the speed of the rocket 3 s after it has left the ground.

After 3 s, the rocket burns out. The motion of the rocket is now modelled as that of a particle moving freely under gravity.

Find the height of the rocket above the ground 5 s after it has left the ground.

A car moves along a horizontal straight road, passing two points A and B. At A the speed of the car is 15 m s^–1. When the driver passes A, he sees a warning sign W ahead of him, 120 m away. He immediately applies the brakes and the car decelerates with uniform deceleration, reaching W with speed 5 m s^–1. At W, the driver sees that the road is clear. He then immediately accelerates the car with uniform acceleration for 16 s to reach a speed of V m s^–1 (V > 15). He then maintains the car at a constant speed of V m s^–1. Moving at this constant speed, the car passes B after a further 22 s.

Sketch, in the space below, a speed-time graph to illustrate the motion of the car as it moves from A to B.

Find the time taken for the car to move from A to B.

The distance from A to B is 1 km.

Find the value of V.

Two particles A and B, of mass m and 2 m respectively, are attached to the ends of a light inextensible string. The particle A lies on a rough horizontal table. The string passes over a small smooth pulley P fixed on the edge of the table. The particle B hangs freely below the pulley, as shown in Figure 3. The coefficient of friction between A and the table is. The particles are released from rest with the string taut. Immediately after release, the magnitude of the acceleration of A and B is . By writing down separate equations of motion for A and B,

find the tension in the string immediately after the particles begin to move,

When B has fallen a distance h, it hits the ground and does not rebound. Particle A is then

Find the speed of A as it reaches P.

State how you have used the information that the string is light.