3.02d Constant acceleration: SUVAT formulae

7 A car of mass 600 kg travels along a straight horizontal road starting from a point \(A\). The resistance to motion of the car is 750 N .

1. The car travels from \(A\) to \(B\) at constant speed in 100 s . The power supplied by the car's engine is constant and equal to 30 kW . Find the distance \(A B\).
2. The car's engine is switched off at \(B\) and the car's speed decreases until the car reaches \(C\) with a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the distance \(B C\).
3. The car's engine is switched on at \(C\) and the power it supplies is constant and equal to 30 kW . The car takes 14 s to travel from \(C\) to \(D\) and reaches \(D\) with a speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the distance \(C D\).

1 An object is released from rest at a height of 125 m above horizontal ground and falls freely under gravity, hitting a moving target \(P\). The target \(P\) is moving on the ground in a straight line, with constant acceleration \(0.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). At the instant the object is released \(P\) passes through a point \(O\) with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the distance from \(O\) to the point where \(P\) is hit by the object.

5 Particle \(P\) travels along a straight line from \(A\) to \(B\) with constant acceleration \(0.05 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Its speed at \(A\) is \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its speed at \(B\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the time taken for \(P\) to travel from \(A\) to \(B\), and find also the distance \(A B\). Particle \(Q\) also travels along the same straight line from \(A\) to \(B\), starting from rest at \(A\). At time \(t \mathrm {~s}\) after leaving \(A\), the speed of \(Q\) is \(k t ^ { 3 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(k\) is a constant. \(Q\) takes the same time to travel from \(A\) to \(B\) as \(P\) does.
2. Find the value of \(k\) and find \(Q\) 's speed at \(B\).

7 A car of mass 1200 kg moves in a straight line along horizontal ground. The resistance to motion of the car is constant and has magnitude 960 N . The car's engine works at a rate of 17280 W .

1. Calculate the acceleration of the car at an instant when its speed is \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car passes through the points \(A\) and \(B\). While the car is moving between \(A\) and \(B\) it has constant speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Show that \(V = 18\). At the instant that the car reaches \(B\) the engine is switched off and subsequently provides no energy. The car continues along the straight line until it comes to rest at the point \(C\). The time taken for the car to travel from \(A\) to \(C\) is 52.5 s .
3. Find the distance \(A C\).

3 A car travels along a straight road with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). It passes through points \(A , B\) and \(C\); the time taken from \(A\) to \(B\) and from \(B\) to \(C\) is 5 s in each case. The speed of the car at \(A\) is \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the distances \(A B\) and \(B C\) are 55 m and 65 m respectively. Find the values of \(a\) and \(u\). [6]

7 \includegraphics[max width=\textwidth, alt={}, center]{631ddcd9-17c0-4a15-8671-40788c3a84d3-3_565_828_1402_660} Particles \(A\) and \(B\) have masses 0.32 kg and 0.48 kg respectively. The particles are attached to the ends of a light inextensible string which passes over a small smooth pulley fixed at the edge of a smooth horizontal table. Particle \(B\) is held at rest on the table at a distance of 1.4 m from the pulley. \(A\) hangs vertically below the pulley at a height of 0.98 m above the floor (see diagram). \(A , B\), the string and the pulley are all in the same vertical plane. \(B\) is released and \(A\) moves downwards.

1. Find the acceleration of \(A\) and the tension in the string. \(A\) hits the floor and \(B\) continues to move towards the pulley. Find the time taken, from the instant that \(B\) is released, for
2. \(A\) to reach the floor,
3. \(B\) to reach the pulley.

3 A cyclist exerts a constant driving force of magnitude \(F \mathrm {~N}\) while moving up a straight hill inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 36 } { 325 }\). A constant resistance to motion of 32 N acts on the cyclist. The total weight of the cyclist and his bicycle is 780 N . The cyclist's acceleration is \(- 0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the value of \(F\). The cyclist's speed is \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the bottom of the hill.
2. Find how far up the hill the cyclist travels before coming to rest.

4 Particles \(P\) and \(Q\) are moving in a straight line on a rough horizontal plane. The frictional forces are the only horizontal forces acting on the particles.

1. Find the deceleration of each of the particles given that the coefficient of friction between \(P\) and the plane is 0.2 , and between \(Q\) and the plane is 0.25 . At a certain instant, \(P\) passes through the point \(A\) and \(Q\) passes through the point \(B\). The distance \(A B\) is 5 m . The velocities of \(P\) and \(Q\) at \(A\) and \(B\) are \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), respectively, both in the direction \(A B\).
2. Find the speeds of \(P\) and \(Q\) immediately before they collide.

6 \includegraphics[max width=\textwidth, alt={}, center]{3e58aa5a-3789-4aaf-8656-b5b98cd7f693-3_518_515_1436_815} Particles \(A\) and \(B\), of masses 0.3 kg and 0.7 kg respectively, are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley. \(A\) is held at rest and \(B\) hangs freely, with both straight parts of the string vertical and both particles at a height of 0.52 m above the floor (see diagram). \(A\) is released and both particles start to move.

1. Find the tension in the string. When both particles are moving with speed \(1.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) the string breaks.
2. Find the time taken, from the instant that the string breaks, for \(A\) to reach the floor. \(7 \quad\) A particle \(P\) starts from rest at a point \(O\) and moves in a straight line. \(P\) has acceleration \(0.6 t \mathrm {~m} \mathrm {~s} ^ { - 2 }\) at time \(t\) seconds after leaving \(O\), until \(t = 10\).
3. Find the velocity and displacement from \(O\) of \(P\) when \(t = 10\). After \(t = 10 , P\) has acceleration \(- 0.4 t \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until it comes to rest at a point \(A\).
4. Find the distance \(O A\).

7 \includegraphics[max width=\textwidth, alt={}, center]{79b90ef5-ef3a-4c59-b662-d0fbfba813ca-4_492_1365_255_392} An elevator is pulled vertically upwards by a cable. The velocity-time graph for the motion is shown above. Find

1. the distance travelled by the elevator,
2. the acceleration during the first stage and the deceleration during the third stage. The mass of the elevator is 800 kg and there is a box of mass 100 kg on the floor of the elevator.
3. Find the tension in the cable in each of the three stages of the motion.
4. Find the greatest and least values of the magnitude of the force exerted on the box by the floor of the elevator.

1 A particle moves up a line of greatest slope of a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = 0.28\). The coefficient of friction between the particle and the plane is \(\frac { 1 } { 3 }\).

1. Show that the acceleration of the particle is \(- 6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
2. Given that the particle's initial speed is \(5.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), find the distance that the particle travels up the plane.

2 \includegraphics[max width=\textwidth, alt={}, center]{fd534430-2619-4078-ad0a-2355e656e121-2_569_519_676_813} Particle \(A\) of mass 0.2 kg and particle \(B\) of mass 0.6 kg are attached to the ends of a light inextensible string. The string passes over a fixed smooth pulley. \(B\) is held at rest at a height of 1.6 m above the floor. \(A\) hangs freely at a height of \(h \mathrm {~m}\) above the floor. Both straight parts of the string are vertical (see diagram). \(B\) is released and both particles start to move. When \(B\) reaches the floor it remains at rest, but \(A\) continues to move vertically upwards until it reaches a height of 3 m above the floor. Find the speed of \(B\) immediately before it hits the floor, and hence find the value of \(h\).

5 A car travels in a straight line from \(A\) to \(B\), a distance of 12 km , taking 552 seconds. The car starts from rest at \(A\) and accelerates for \(T _ { 1 } \mathrm {~s}\) at \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), reaching a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car then continues to move at \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for \(T _ { 2 } \mathrm {~s}\). It then decelerates for \(T _ { 3 } \mathrm {~s}\) at \(1 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), coming to rest at \(B\).

1. Sketch the velocity-time graph for the motion and express \(T _ { 1 }\) and \(T _ { 3 }\) in terms of \(V\).
2. Express the total distance travelled in terms of \(V\) and show that \(13 V ^ { 2 } - 3312 V + 72000 = 0\). Hence find the value of \(V\).

4 Particles \(P\) and \(Q\) move on a straight line \(A O B\). The particles leave \(O\) simultaneously, with \(P\) moving towards \(A\) and with \(Q\) moving towards \(B\). The initial speed of \(P\) is \(1.3 \mathrm {~ms} ^ { - 1 }\) and its acceleration in the direction \(O A\) is \(0.1 \mathrm {~m} \mathrm {~s} ^ { - 2 } . Q\) moves with acceleration in the direction \(O B\) of \(0.016 t \mathrm {~m} \mathrm {~s} ^ { - 2 }\), where \(t\) seconds is the time elapsed since the instant that \(P\) and \(Q\) started to move from \(O\). When \(t = 20\), particle \(P\) passes through \(A\) and particle \(Q\) passes through \(B\).

1. Given that the speed of \(Q\) at \(B\) is the same as the speed of \(P\) at \(A\), find the speed of \(Q\) at time \(t = 0\).
2. Find the distance \(A B\).

6 A particle of mass 3 kg falls from rest at a point 5 m above the surface of a liquid which is in a container. There is no instantaneous change in speed of the particle as it enters the liquid. The depth of the liquid in the container is 4 m . The downward acceleration of the particle while it is moving in the liquid is \(5.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Find the resistance to motion of the particle while it is moving in the liquid.
2. Sketch the velocity-time graph for the motion of the particle, from the time it starts to move until the time it reaches the bottom of the container. Show on your sketch the velocity and the time when the particle enters the liquid, and when the particle reaches the bottom of the container.

1 A particle \(P\) is projected vertically upwards with speed \(11 \mathrm {~ms} ^ { - 1 }\) from a point on horizontal ground. At the same instant a particle \(Q\) is released from rest at a point \(h \mathrm {~m}\) above the ground. \(P\) and \(Q\) hit the ground at the same instant, when \(Q\) has speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the time after projection at which \(P\) hits the ground.
2. Hence find the values of \(h\) and \(V\).

2 A particle of mass 0.5 kg starts from rest and slides down a line of greatest slope of a smooth plane. The plane is inclined at an angle of \(30 ^ { \circ }\) to the horizontal.

1. Find the time taken for the particle to reach a speed of \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the particle has travelled 3 m down the slope from its starting point, it reaches rough horizontal ground at the bottom of the slope. The frictional force acting on the particle is 1 N .
2. Find the distance that the particle travels along the ground before it comes to rest.

3 A lorry of mass 24000 kg is travelling up a hill which is inclined at \(3 ^ { \circ }\) to the horizontal. The power developed by the lorry's engine is constant, and there is a constant resistance to motion of 3200 N .

1. When the speed of the lorry is \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), its acceleration is \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the power developed by the lorry's engine.
2. Find the steady speed at which the lorry moves up the hill if the power is 500 kW and the resistance remains 3200 N .

7 A cyclist starts from rest at point \(A\) and moves in a straight line with acceleration \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for a distance of 36 m . The cyclist then travels at constant speed for 25 s before slowing down, with constant deceleration, to come to rest at point \(B\). The distance \(A B\) is 210 m .

1. Find the total time that the cyclist takes to travel from \(A\) to \(B\). 24 s after the cyclist leaves point \(A\), a car starts from rest from point \(A\), with constant acceleration \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), towards \(B\). It is given that the car overtakes the cyclist while the cyclist is moving with constant speed.
2. Find the time that it takes from when the cyclist starts until the car overtakes her.

2 A particle is released from rest at a point \(H \mathrm {~m}\) above horizontal ground and falls vertically. The particle passes through a point 35 m above the ground with a speed of \(( V - 10 ) \mathrm { m } \mathrm { s } ^ { - 1 }\) and reaches the ground with a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find

1. the value of \(V\),
2. the value of \(H\).

5 \includegraphics[max width=\textwidth, alt={}, center]{48f66bd5-33c1-4ce9-85f9-69faf10e871c-3_259_828_1288_660} A smooth inclined plane of length 2.5 m is fixed with one end on the horizontal floor and the other end at a height of 0.7 m above the floor. Particles \(P\) and \(Q\), of masses 0.5 kg and 0.1 kg respectively, are attached to the ends of a light inextensible string which passes over a small smooth pulley fixed at the top of the plane. Particle \(Q\) is held at rest on the floor vertically below the pulley. The string is taut and \(P\) is at rest on the plane (see diagram). \(Q\) is released and starts to move vertically upwards towards the pulley and \(P\) moves down the plane.

1. Find the tension in the string and the magnitude of the acceleration of the particles before \(Q\) reaches the pulley. At the instant just before \(Q\) reaches the pulley the string breaks; \(P\) continues to move down the plane and reaches the floor with a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the length of the string.

4 \includegraphics[max width=\textwidth, alt={}, center]{f23ea8e7-9b81-4192-8c20-8c46aabfecca-3_442_495_255_826} Particles \(A\) and \(B\), of masses 0.35 kg and 0.15 kg respectively, are attached to the ends of a light inextensible string which passes over a fixed smooth pulley. The system is at rest with \(B\) held on the horizontal floor, the string taut and its straight parts vertical. \(A\) is at a height of 1.6 m above the floor (see diagram). \(B\) is released and the system begins to move; \(B\) does not reach the pulley. Find

1. the acceleration of the particles and the tension in the string before \(A\) reaches the floor,
2. the greatest height above the floor reached by \(B\).

5 A cyclist and his bicycle have a total mass of 90 kg . The cyclist starts to move with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from the top of a straight hill, of length 500 m , which is inclined at an angle of \(\sin ^ { - 1 } 0.05\) to the horizontal. The cyclist moves with constant acceleration until he reaches the bottom of the hill with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The cyclist generates 420 W of power while moving down the hill. The resistance to the motion of the cyclist and his bicycle, \(R \mathrm {~N}\), and the cyclist's speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), both vary.

1. Show that \(R = \frac { 420 } { v } + 43.56\).
2. Find the cyclist's speed at the mid-point of the hill. Hence find the decrease in the value of \(R\) when the cyclist moves from the top of the hill to the mid-point of the hill, and when the cyclist moves from the mid-point of the hill to the bottom of the hill.

7 A car starts from rest and moves in a straight line from point \(A\) with constant acceleration \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 10 s . The car then travels at constant speed for 30 s before decelerating uniformly, coming to rest at point \(B\). The distance \(A B\) is 1.5 km .

1. Find the total distance travelled in the first 40 s of motion. When the car has been moving for 20 s , a motorcycle starts from rest and accelerates uniformly in a straight line from point \(A\) to a speed \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). It then maintains this speed for 30 s before decelerating uniformly to rest at point \(B\). The motorcycle comes to rest at the same time as the car.
2. Given that the magnitude of the acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) of the motorcycle is three times the magnitude of its deceleration, find the value of \(a\).
3. Sketch the displacement-time graph for the motion of the car.

3 \includegraphics[max width=\textwidth, alt={}, center]{94c11160-a718-4de5-867a-27c755051fa6-2_312_1207_1320_468} Particles \(P\) and \(Q\), of masses 7 kg and 3 kg respectively, are attached to the two ends of a light inextensible string. The string passes over two small smooth pulleys attached to the two ends of a horizontal table. The two particles hang vertically below the two pulleys. The two particles are both initially at rest, 0.5 m below the level of the table, and 0.4 m above the horizontal floor (see diagram).

1. Find the acceleration of the particles and the speed of \(P\) immediately before it reaches the floor.
2. Determine whether \(Q\) comes to instantaneous rest before it reaches the pulley directly above it.