3.02d Constant acceleration: SUVAT formulae

5. A car and a motorbike are at rest adjacent to one another at a set of traffic lights on a long, straight stretch of road. They set off simultaneously at time \(t = 0\). The motorcyclist accelerates uniformly at \(6 \mathrm {~ms} ^ { - 2 }\) until he reaches a speed of \(30 \mathrm {~ms} ^ { - 1 }\) which he then maintains. The car driver accelerates uniformly for 9 seconds until she reaches \(36 \mathrm {~ms} ^ { - 1 }\) and then remains at this speed.

1. Find the acceleration of the car.
2. Draw on the same diagram speed-time graphs to illustrate the movements of both vehicles.
3. Find the value of \(t\) when the car again draws level with the motorcyclist.

2. An underground train accelerates uniformly from rest at station \(A\) to a velocity of \(24 \mathrm {~ms} ^ { - 1 }\). It maintains this speed for 84 seconds, until it decelerates uniformly to rest at station \(B\). The total journey time is 116 seconds and the magnitudes of the acceleration and deceleration are equal.

1. Find the time it takes the train to accelerate from rest to \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Illustrate this information on a velocity-time graph.
3. Using your graph, or otherwise, find the distance between the two stations.

6. A boy kicks a football vertically upwards from a height of 0.6 m above the ground with a speed of \(10.5 \mathrm {~ms} ^ { - 1 }\). The ball is modelled as a particle and air resistance is ignored.

1. Find the greatest height above the ground reached by the ball.
2. Calculate the length of time for which the ball is more than 2 m above the ground.

2. During trials of a bullet-proof vest, a shotgun of mass 2 kg is used to fire a bullet of mass 30 g horizontally at the vest. The initial speed of the bullet is \(100 \mathrm {~ms} ^ { - 1 }\).

1. Calculate the initial speed of recoil of the gun. The bullet hits the vest horizontally at a speed of \(80 \mathrm {~ms} ^ { - 1 }\) and is brought uniformly to rest in a distance of 2 cm .
2. Find the magnitude of the force exerted by the vest on the bullet in bringing it to rest.
   (4 marks)

6. A student attempts to sketch the acceleration-time graph of a parachutist who jumps from a plane at a height of 2200 m above the ground. The student assumes that the parachutist falls freely from rest under gravity until she is 240 m from the ground at which point she opens her parachute. The student makes the assumption that, at this point, the velocity of the parachutist is immediately reduced to a value which remains constant until she reaches the ground 140 seconds after she left the plane. \begin{figure}[h] \end{figure} The student decides to ignore air resistance and his sketch is shown in Figure 3. The value \(t _ { 1 }\) is used by the student to denote the time at which the parachute is opened. Using the model proposed by the student, calculate

1. the speed of the parachutist immediately before she opens her parachute,
2. the value of \(t _ { 1 }\),
3. the speed of the parachutist after the parachute is opened.
4. Comment on two features of the student's model which are unrealistic and say what effect taking account of these would have had on the values which you calculated in parts (a) and (b).
   (4 marks)

7. A machine fires ball-bearings up the line of greatest slope of a rough plane inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 3 } { 5 }\). The coefficient of friction between the ball-bearings and the plane is \(\frac { 1 } { 4 }\).

1. Show that the magnitude of the acceleration of the ball-bearings is \(\frac { 4 } { 5 } g\) and state its direction. Given that the machine is placed at a point \(A , 30 \mathrm {~m}\) from the top edge of the plane, and the ball-bearings are projected with an initial speed of \(20 \mathrm {~ms} ^ { - 1 }\),
2. find, giving your answer to the nearest cm , how close the ball-bearings get to the top edge of the plane.
3. How long does it take for a ball-bearing to travel from the highest point it reaches back down to the point \(A\) again? END

4. A bullet of mass 50 g is fired horizontally at a wooden block of mass 4.95 kg which is lying at rest on a rough horizontal surface. The bullet enters the block at \(400 \mathrm {~ms} ^ { - 1 }\) and becomes embedded in the block.

1. Find the speed with which the block begins to move. Given that the block decelerates uniformly to rest over a distance of 4 m ,
2. show that the coefficient of friction is \(\frac { 2 } { g }\).

6. A particle moving in a straight line with speed \(5 U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) undergoes a uniform deceleration for 6 seconds which reduces its speed to \(2 \mathrm { Um } \mathrm { s } ^ { - 1 }\). It maintains this speed for 16 seconds before uniformly decelerating to rest in a further 2 seconds.

1. Sketch a speed-time graph displaying this information.
2. Find an expression for each of the decelerations in terms of \(U\). Given that the total distance travelled by the particle during this period of motion is 220 m ,
3. find the value of \(U\).

7. A car of mass 1200 kg tows a trailer of mass 800 kg along a straight level road by means of a rigid towbar. The resistances to the motion of the car and the trailer are proportional to their masses. Given that the car experiences a resistance to motion of 300 N ,

1. find the resistance to motion which the trailer experiences. Given that the engine of the car exerts a driving force of 3 kN ,
2. find the acceleration of the system,
3. show that the tension in the towbar is 1200 N . When the system has reached a speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it continues at constant speed until an electrical fault causes the engine of the car to switch off. The brakes are used to apply a constant retarding force until the system comes to rest. Given that the retarding force of the brakes has magnitude 1 kN and assuming that the original resistances to motion of the car and the trailer remain constant,
4. calculate the distance that the system travels during the braking period,
5. find the magnitude and direction of the force exerted by the towbar on the car.
6. Comment on the assumption that the original resistances to motion of the car and the trailer remain constant throughout the motion.

3. A lorry accelerates uniformly from \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 30 seconds.

1. Find how far it travels while accelerating.
2. Find, in seconds correct to 2 decimal places, the length of time it takes for the lorry to cover the first half of this distance.
   (6 marks)

5. A sledgehammer of mass 12 kg is being used to drive a wooden post of mass 4 kg into the ground. A labourer moves the sledgehammer from rest at a point 0.5 m vertically above the post with constant acceleration \(16 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) directed towards the post.

1. Find the velocity with which the sledgehammer hits the post. When the sledgehammer hits the post, they both move together with common speed, \(V\).
2. Show that \(V = 3 \mathrm {~ms} ^ { - 1 }\). As the sledgehammer hits the post, the labourer relaxes his grip and applies no further force. The sledgehammer and post are brought to rest by the action of a resistive force from the ground of magnitude 1500 N .
3. Find, in centimetres, the total distance that the sledgehammer and the post travel together before coming to rest.

5. A car on a straight test track starts from rest and accelerates to a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 6 seconds. The car maintains this speed for a further 50 seconds before decelerating to rest. In a simple model of this motion, the acceleration and deceleration are assumed to be uniform and the magnitude of the deceleration to be 1.5 times that of the acceleration.

1. Show that the total time for which the car is moving is 60 seconds.
2. Sketch a velocity-time graph for this journey. Given that the total distance travelled is 1320 metres,
3. find \(V\). In a more sophisticated model, the acceleration is assumed to be inversely proportional to the velocity of the car.
4. Explain how the acceleration would vary during the first six seconds under this model.
   (2 marks)

7. A car of mass 1250 kg tows a caravan of mass 850 kg up a hill inclined at an angle \(\alpha\) to the horizontal where \(\sin \alpha = \frac { 1 } { 14 }\). The total resistance to motion experienced by the car is 400 N , and by the caravan is 500 N . Given that the driving force of the engine is 3 kN ,

1. show that the acceleration of the system is \(0.3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\),
2. find the tension in the towbar linking the car and the caravan. Starting from rest, the car accelerates uniformly for 540 m until it reaches a speed of \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the top of the hill.
3. Find v. At the top of the hill the road becomes level and the driver maintains the speed at which the car and caravan reached the top of the hill.
4. Assuming that the resistance to motion on each part of the system is unchanged, find the percentage reduction in the driving force of the engine required to achieve this.

3. A small van of mass 1500 kg is used to tow a car of mass 750 kg by means of a rope of length 9 m joined to both vehicles. The van sets off with the rope slack and reaches a speed of \(2 \mathrm {~ms} ^ { - 1 }\) just before the rope becomes taut and jerks the car into motion. Immediately after the rope becomes taut, the van and car travel with common speed \(V \mathrm {~ms} ^ { - 1 }\).

1. Show that \(V = \frac { 4 } { 3 }\).
2. Calculate the magnitude of the impulse on the car when the rope tightens. The van and car eventually reach a steady speed of \(18 \mathrm {~ms} ^ { - 1 }\) with the rope taut when a child runs out into the road, 30 m in front of the van. The van driver brakes sharply and decelerates uniformly to rest in a distance of 27 m . It takes the driver of the car 1 second to react to the van starting to brake. He then brakes and the car decelerates uniformly at \(f \mathrm {~m} \mathrm {~s} ^ { - 2 }\), coming to rest before colliding with the van.
3. Find the set of possible values of \(f\).
   (5 marks)

5. Two flies \(P\) and \(Q\), are crawling vertically up a wall. At time \(t = 0\), the flies are at the same height above the ground, with \(P\) crawling at a steady speed of \(4 \mathrm { cms } ^ { - 1 }\). \(Q\) starts from rest at time \(t = 0\) and accelerates uniformly to a speed of \(6 \mathrm {~cm} \mathrm {~s} ^ { - 1 }\) in 6 seconds. Fly \(Q\) then maintains this speed.

1. Find the value of \(t\) when the two flies are moving at the same speed.
2. Sketch on the same diagram, speed-time graphs to illustrate the motion of the two flies. Given that the distance of the two flies from the top of the wall at time \(t = 0\) is \(x \mathrm {~cm}\) and that \(Q\) reaches the top of the wall first,
3. show that \(x > 36\).

4 A point P on a piece of machinery is moving in a vertical straight line. The displacement of P above ground level at time \(t\) seconds is \(y\) metres. The displacement-time graph for the motion during the time interval \(0 \leqslant t \leqslant 4\) is shown in Fig. 7 . \begin{figure}[h] \end{figure}

1. Using the graph, determine for the time interval \(0 \leqslant t \leqslant 4\) (A) the greatest displacement of P above its position when \(t = 0\),
   (B) the greatest distance of P from its position when \(t = 0\),
   (C) the time interval in which P is moving downwards,
   (D) the times when P is instantaneously at rest. The displacement of P in the time interval \(0 \leqslant t \leqslant 3\) is given by \(y = - 4 t ^ { 2 } + 8 t + 12\).
2. Use calculus to find expressions in terms of \(t\) for the velocity and for the acceleration of P in the interval \(0 \leqslant t \leqslant 3\).
3. At what times does P have a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the interval \(0 \leqslant t \leqslant 3\) ? In the time interval \(3 \leqslant t \leqslant 4 , \mathrm { P }\) has a constant acceleration of \(32 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). There is no sudden change in velocity when \(t = 3\).
4. Find an expression in terms of \(t\) for the displacement of P in the interval \(3 \leqslant t \leqslant 4\).

1 Fig. 4 illustrates a straight horizontal road. \(A\) and \(B\) are points on the road which are 215 metres apart and \(M\) is the mid-point of AB . When a car passes A its speed is \(12 \mathrm {~ms} ^ { - 1 }\) in the direction AB . It then accelerates uniformly and when it reaches \(B\) its speed is \(31 \mathrm {~ms} ^ { - 1 }\). \begin{figure}[h] \end{figure}

1. Find the car's acceleration.
2. Find how long it takes the car to travel from A to B .
3. Find how long it takes the car to travel from A to M .
4. Explain briefly, in terms of the speed of the car, why the time taken to travel from A to M is more than half the time taken to travel from A to B .

4 Fig. 4 illustrates points \(\mathrm { A } , \mathrm { B }\) and C on a straight race track. The distance AB is 300 m and AC is 500 m .
A car is travelling along the track with uniform acceleration. \begin{figure}[h] \end{figure} Initially the car is at A and travelling in the direction AB with speed \(5 \mathrm {~ms} ^ { - 1 }\). After 20s it is at C .

1. Find the acceleration of the car.
2. Find the speed of the car at B and how long it takes to travel from A to B .

5 A particle is moving along a straight line and its position is relative to an origin on the line. At time \(t \mathrm {~s}\), the particle's acceleration, \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\), is given by $$a = 6 t - 12 .$$ At \(t = 0\) the velocity of the particle is \(+ 9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and its position is - 2 m .

1. Find an expression for the velocity of the particle at time \(t \mathrm {~s}\) and verify that it is stationary when \(t = 3\).
2. Find the position of the particle when \(t = 2\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{b9e41fac-9f4b-4165-af03-67ebdcb326de-3_349_987_375_623} \captionsetup{labelformat=empty} \caption{Fig. 4}
   \end{figure} Particles P and Q move in the same straight line. Particle P starts from rest and has a constant acceleration towards \(Q\) of \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Particle \(Q\) starts 125 m from \(P\) at the same time and has a constant speed of \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) away from \(P\). The initial values are shown in Fig. 4.

1. Write down expressions for the distances travelled by P and by Q at time \(t\) seconds after the start of the motion.
2. How much time does it take for P to catch up with Q and how far does P travel in this time?

2 Fig. 3 shows two people, Sam and Tom, pushing a car of mass 1000 kg along a straight line \(l\) on level ground. Sam pushes with a constant horizontal force of 300 N at an angle of \(30 ^ { \circ }\) to the line \(l\).
Tom pushes with a constant horizontal force of 175 N at an angle of \(15 ^ { \circ }\) to the line \(l\). \begin{figure}[h] \end{figure}

1. The car starts at rest and moves with constant acceleration. After 6 seconds it has travelled 7.2 m . Find its acceleration.
2. Find the resistance force acting on the car along the line \(l\).
3. The resultant of the forces exerted by Sam and Tom is not in the direction of the car's acceleration. Explain briefly why.

3 A particle is travelling along a straight line with constant acceleration. \(\mathrm { P } , \mathrm { O }\) and Q are points on the line, as illustrated in Fig. 4. The distance from P to O is 5 m and the distance from O to Q is 30 m . \begin{figure}[h] \end{figure} Initially the particle is at O . After 10 s , it is at Q and its velocity is \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the direction \(\overrightarrow { \mathrm { OQ } }\).

1. Find the initial velocity and the acceleration of the particle.
2. Prove that the particle is never at P .

4 A car is driven with constant acceleration, \(a \mathrm {~m} \mathrm {~s} { } ^ { 2 }\), along a straight road. Its speed when it passes a road sign is \(u \mathrm {~ms} { } ^ { 1 }\). The car travels 14 m in the 2 seconds after passing the sign; 5 seconds after passing the sign it has a speed of \(19 \mathrm {~ms} { } ^ { 1 }\).

1. Write down two equations connecting \(a\) and \(u\). Hence find the values of \(a\) and \(u\).
2. What distance does the car travel in the 5 seconds after passing the road sign?

4 A ring is moving on a straight wire. Its velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds after passing a point Q .
Model A for the motion of the ring gives the velocity-time graph for \(0 \leqslant t \leqslant 6\) shown in Fig. 7 . \begin{figure}[h] \end{figure} Use model A to calculate the following.

1. The acceleration of the ring when \(t = 0.5\).
2. The displacement of the ring from Q when
   (A) \(t = 2\),
   (B) \(t = 6\). In an alternative model B , the velocity of the ring is given by \(v = 2 t ^ { 2 } - 14 t + 20\) for \(0 \leqslant t \leqslant 6\).
3. Calculate the acceleration of the ring at \(t = 0.5\) as given by model B.
4. Calculate by how much the models differ in their values for the least \(v\) in the time interval \(0 \leqslant t \leqslant 6\).
5. Calculate the displacement of the ring from Q when \(t = 6\) as given by model B .