3.02d Constant acceleration: SUVAT formulae

8. Two trams, tram \(A\) and tram \(B\), run on parallel straight horizontal tracks. Initially the two trams are at rest in the depot and level with each other. At time \(t = 0 , \operatorname { tram } A\) starts to move. Tram \(A\) moves with constant acceleration \(2 \mathrm {~ms} ^ { - 2 }\) for 5 seconds and then continues to move along the track at constant speed. At time \(t = 20\) seconds, tram \(B\) starts from rest and moves in the same direction as tram \(A\). Tram \(B\) moves with constant acceleration \(3 \mathrm {~ms} ^ { - 2 }\) for 4 seconds and then continues to move along the track at constant speed. The trams are modelled as particles.

1. Sketch, on the same axes, a speed-time graph for the motion of tram \(A\) and a speed-time graph for the motion of tram \(B\), from \(t = 0\) to the instant when tram \(B\) overtakes \(\operatorname { tram } A\). At the instant when the two trams are moving with the same speed, \(\operatorname { tram } A\) is \(d\) metres in front of tram \(B\).
2. Find the value of \(d\).
3. Find the distance of the trams from the depot at the instant when tram \(B\) overtakes \(\operatorname { tram } A\). \includegraphics[max width=\textwidth, alt={}, center]{5a2cf693-d966-4787-8778-ecc8a79a6265-32_2647_1835_118_116}
   VALV SIHI NI IIIIIM ION OC
   VALV SIHI NI IMIMM ION OO
   VIUV SIHI NI JIIXM ION OC

2. A motorbike is moving with constant acceleration along a straight horizontal road. The motorbike passes a point \(P\) and 10 seconds later passes a point \(Q\). The speed of the motorbike as it passes \(Q\) is \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) Given that \(P Q = 220 \mathrm {~m}\),

1. find the acceleration of the motorbike,
2. find the distance travelled by the motorbike during the fifth second after passing \(P\) VILV SIHI NI IIII M I ON OC
   VARV SIHI NI JLIUMI ON OC
   VIIV SIHIL NI IMINM ION OC

7. Two small children, Ajaz and Beth, are running a 100 m race along a straight horizontal track. They both start from rest, leaving the start line at the same time. Ajaz accelerates at \(0.8 \mathrm {~ms} ^ { - 2 }\) up to a speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and then maintains this speed until he crosses the finish line. Beth accelerates at \(1 \mathrm {~ms} ^ { - 2 }\) for \(T\) seconds and then maintains a constant speed until she crosses the finish line. Ajaz and Beth cross the finish line at the same time.

1. Sketch, on the same axes, a speed-time graph for each child, from the instant when they leave the start line to the instant when they cross the finish line.
2. Find the time taken by Ajaz to complete the race.
3. Find the value of \(T\)
4. Find the difference in the speeds of the two children as they cross the finish line.

1. Two students observe a book of mass 0.2 kg fall vertically from rest from a shelf that is 1.5 m above the floor.

Student \(A\) suggests that the book is modelled as a particle falling freely under gravity.

1. Use student \(A\) 's model to find the time taken for the book to reach the floor. Student \(B\) suggests an improved model where the book is modelled as a particle experiencing a constant resistance to motion of magnitude \(R\) newtons. Given that the time taken for the book to reach the floor is 0.6 seconds,
2. use student \(B\) 's model to find the value of \(R\)

5. \begin{figure}[h] \end{figure} The speed-time graph in Figure 2 illustrates the motion of a car travelling along a straight horizontal road.
At time \(t = 0\), the car starts from rest and accelerates uniformly for 30 s until it reaches a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The car then travels at a constant speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) until time \(t = T\) seconds.

1. Show that the distance travelled by the car between \(t = 0\) and \(t = T\) seconds is \(V ( T - 15 )\) metres. A motorbike also travels along the same road.
   At time \(t = T\) seconds, the distance travelled by each vehicle is the same.
2. Find the value of \(T\) \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{f2737a11-4a15-41e9-9f87-31a705a8948b-15_643_1266_1882_402} \captionsetup{labelformat=empty} \caption{Figure 2}
   \end{figure}

7. \begin{figure}[h] \end{figure} A car of mass 1200 kg is towing a trailer of mass 600 kg up a straight road, as shown in Figure 4. The road is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 12 }\) The driving force produced by the engine of the car is 3000 N .
The car moves with acceleration \(0.75 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) The non-gravitational resistance to motion of

• the car is modelled as a constant force of magnitude \(2 R\) newtons
• the trailer is modelled as a constant force of magnitude \(R\) newtons

The car and the trailer are modelled as particles.
The tow bar between the car and trailer is modelled as a light rod that is parallel to the direction of motion. Using the model,

1. show that the value of \(R\) is 60
2. find the tension in the tow bar. When the car and trailer are moving at a speed of \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the tow bar breaks.
   Given that the non-gravitational resistance to motion of the trailer remains unchanged,
3. use the model to find the further distance moved by the trailer before it first comes to rest.

1. A parachute is used to deliver a box of supplies. The parachute is attached to the box.

• the parachute and box are dropped from rest from a helicopter that is hovering at a height of 520 m above the ground
• the parachute and box fall vertically and freely under gravity for 5 seconds, then the parachute opens
• from the instant the parachute opens, it provides a resistance to motion of magnitude 3200 N
• the parachute and box continue to fall vertically downwards after the parachute opens
• the parachute and box are modelled throughout the motion as a particle \(P\) of mass 250 kg
  1. Find the distance fallen by \(P\) in the first 5 seconds.
  2. Find the speed with which \(P\) lands on the ground.
  3. Find the total time from the instant when \(P\) is dropped from the helicopter to the instant when \(P\) lands on the ground.
  4. Sketch a speed-time graph for the motion of \(P\) from the instant when \(P\) is dropped from the helicopter to the instant when \(P\) lands on the ground.

6. \begin{figure}[h] \end{figure} A box of mass \(m\) lies on a rough horizontal plane. The box is pulled along the plane in a straight line at constant speed by a light rope. The rope is inclined at an angle \(\theta\) to the plane, as shown in Figure 3.
The coefficient of friction between the box and the plane is \(\frac { 1 } { 3 }\) The box is modelled as a particle.
Given that \(\tan \theta = \frac { 3 } { 4 }\)

1. find, in terms of \(m\) and \(g\), the tension in the rope. The rope is now removed and the box is placed at rest on the plane.
   The box is then projected horizontally along the plane with speed \(u\).
   The box is again modelled as a particle.
   When the box has moved a distance \(d\) along the plane, the speed of the box is \(\frac { 1 } { 2 } u\).
2. Find \(d\) in terms of \(u\) and \(g\).

   VJYV SIHI NI JIIIM ION OC

   vauv sthin NI JLHMA LON OO

   V34V SIHI NI IIIIM ION OC

6. Two cars \(A\) and \(B\) are moving in the same direction along a straight horizontal road. Car \(A\) is moving with uniform acceleration \(0.4 \mathrm {~ms} ^ { - 2 }\) and car \(B\) is moving with uniform acceleration \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). At the instant when \(B\) is 200 m behind \(A\), the speed of \(A\) is \(35 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the speed of \(B\) is \(44 \mathrm {~ms} ^ { - 1 }\). Find the speed of \(B\) when it overtakes \(A\).
(9)

7. A train moves on a straight horizontal track between two stations \(A\) and \(B\). The train starts from rest at \(A\) and moves with constant acceleration \(1 \mathrm {~ms} ^ { - 2 }\) until it reaches a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The train maintains this speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for the next \(T\) seconds before slowing down with constant deceleration \(0.5 \mathrm {~ms} ^ { - 2 }\), coming to rest at \(B\). The journey from \(A\) to \(B\) takes 180 s and the distance between the stations is 4800 m .

1. Sketch a speed-time graph for the motion of the train from \(A\) to \(B\).
2. Show that \(T = 180 - 3 V\).
3. Find the value of \(V\).

8. \begin{figure}[h] \end{figure} Two particles \(P\) and \(Q\) have masses 2 kg and 3 kg respectively. The particles are attached to the ends of a light inextensible string. The string passes over a smooth light pulley which is fixed at the top of a rough plane. The plane is inclined to horizontal ground at an angle \(\alpha\), where tan \(\alpha = \frac { 3 } { 4 }\). Initially \(P\) is held at rest on the inclined plane with the part of the string from \(P\) to the pulley parallel to a line of greatest slope of the plane. The particle \(Q\) hangs freely below the pulley at a height of 0.5 m above the ground, as shown in Figure 3. The coefficient of friction between \(P\) and the plane is \(\mu\). The system is released from rest, with the string taut, and \(Q\) strikes the ground before \(P\) reaches the pulley. The speed of \(Q\) at the instant when it strikes the ground is \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. For the motion before \(Q\) strikes the ground, find the tension in the string.
2. Find the value of \(\mu\).

   END

1. At time \(t = 0\), a stone is thrown vertically upwards with speed \(19.6 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(A\) which is \(h\) metres above horizontal ground. At time \(t = 3 \mathrm {~s}\), another stone is released from rest from a point \(B\) which is also \(h\) metres above the same horizontal ground. Both stones hit the ground at time \(t = T\) seconds. The motion of each stone is modelled as that of a particle moving freely under gravity.

Find

1. the value of \(T\),
2. the value of \(h\).
   

   VILU SIHI NI III M I ION OC

   VIIV 5141 NI JINAM ION OC

   VI4V SIHI NI JIIYM ION OO

4. \begin{figure}[h] \end{figure} A particle \(P\) of mass \(m \mathrm {~kg}\) is attached to one end of a light inextensible string of length 2.5 m . The other end of the string is attached to a fixed point \(A\) on a vertical wall. The tension in the string is 16 N . The particle is held in equilibrium by a force of magnitude \(F\) newtons, acting in the vertical plane which is perpendicular to the wall and contains the string. This force acts in a direction perpendicular to the string, as shown in Figure 2. Given that the horizontal distance of \(P\) from the wall is 1.5 m , find

1. the value of \(F\),
2. the value of \(m\).
   \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5f2d38d9-b719-4205-8cb0-caa959afc46f-16_186_830_292_557} \captionsetup{labelformat=empty} \caption{Figure 3}
   \end{figure} Two posts, \(A\) and \(B\), are fixed at the side of a straight horizontal road and are 816 m apart, as shown in Figure 3. A car and a van are at rest side by side on the road and level with \(A\). The car and the van start to move at the same time in the direction \(A B\). The car accelerates from rest with constant acceleration until it reaches a speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car then moves at a constant speed of \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The van accelerates from rest with constant acceleration for 12 s until it reaches a speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The van then moves at a constant speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When the car has been moving at \(24 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for 30 s , the van draws level with the car at \(B\), and each vehicle has then travelled a distance of 816 m .
   1. Sketch, on the same diagram, a speed-time graph for the motion of each vehicle from \(A\) to \(B\).
   2. Find the time for which the car is accelerating.
   3. Find the value of \(V\).

8. \begin{figure}[h] \end{figure} A rough plane is inclined at \(30 ^ { \circ }\) to the horizontal. A particle \(P\) of mass 0.5 kg is held at rest on the plane by a horizontal force of magnitude 5 N , as shown in Figure 4. The force acts in a vertical plane containing a line of greatest slope of the inclined plane. The particle is on the point of moving up the plane.

1. Find the magnitude of the normal reaction of the plane on \(P\).
2. Find the coefficient of friction between \(P\) and the plane. The force of magnitude 5 N is now removed and \(P\) accelerates from rest down the plane.
3. Find the speed of \(P\) after it has travelled 3 m down the plane.

3. A car is moving at a constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) along a straight horizontal road. The car is modelled as a particle.
At time \(t = 0\), the car is at the point \(A\) and the driver sees a road sign 48 m ahead.
Let \(t\) seconds be the time that elapses after the car passes \(A\).
In a first model, the car is assumed to decelerate uniformly at \(6 \mathrm {~ms} ^ { - 2 }\) from \(A\) until the car reaches the road sign.

1. Use this first model to find the speed of the car as it reaches the sign. The road sign indicates that the speed limit immediately after the sign is \(13 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   In a second model, the car is assumed to decelerate uniformly at \(6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) from \(A\) until it reaches a speed of \(13 \mathrm {~ms} ^ { - 1 }\). The car then maintains this speed until it reaches the road sign.
2. Use this second model to find the value of \(t\) at which the car reaches the sign. In a third model, the car is assumed to move with constant speed \(25 \mathrm {~ms} ^ { - 1 }\) from \(A\) until time \(t = 0.2\), the car then decelerates uniformly at \(6 \mathrm {~ms} ^ { - 2 }\) until it reaches a speed of \(13 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car then maintains this speed until it reaches the road sign.
3. Use this third model to find the value of \(t\) at which the car reaches the sign.

2. \begin{figure}[h] \end{figure} Two fixed points, \(A\) and \(B\), are on a straight horizontal road.
The acceleration-time graph in Figure 2 represents the motion of a car travelling along the road as it moves from \(A\) to \(B\). At time \(t = 0\), the car passes through \(A\) with speed \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) At time \(t = 20 \mathrm {~s}\), the car passes through \(B\) with speed \(v \mathrm {~ms} ^ { - 1 }\)

1. Show that \(v = 18\)
2. Sketch a speed-time graph for the motion of the car from \(A\) to \(B\).
3. Find the distance \(A B\).

6. \begin{figure}[h] \end{figure} Two cars, \(A\) and \(B\), move on parallel straight horizontal tracks. Initially \(A\) and \(B\) are both at rest with \(A\) at the point \(P\) and \(B\) at the point \(Q\), as shown in Figure 2. At time \(t = 0\) seconds, \(A\) starts to move with constant acceleration \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\) for 3.5 s , reaching a speed of \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Car \(A\) then moves with constant speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Find the value of \(a\). Car \(B\) also starts to move at time \(t = 0\) seconds, in the same direction as car \(A\). Car \(B\) moves with a constant acceleration of \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). At time \(t = T\) seconds, \(B\) overtakes \(A\). At this instant \(A\) is moving with constant speed.
2. On a diagram, sketch, on the same axes, a speed-time graph for the motion of \(A\) for the interval \(0 \leqslant t \leqslant T\) and a speed-time graph for the motion of \(B\) for the interval \(0 \leqslant t \leqslant T\).
3. Find the value of \(T\).
4. Find the distance of car \(B\) from the point \(Q\) when \(B\) overtakes \(A\).
5. On a new diagram, sketch, on the same axes, an acceleration-time graph for the motion of \(A\) for the interval \(0 \leqslant t \leqslant T\) and an acceleration-time graph for the motion of \(B\) for the interval \(0 \leqslant t \leqslant T\). \(\_\_\_\_\) VAYV SIHI NI JIIIM ION OC
   VJYV SIHI NI JIIIM ION OC
   VJYV SIHI NI JLIYM ION OC

6. A parachutist drops from a helicopter \(H\) and falls vertically from rest towards the ground. Her parachute opens 2 s after she leaves \(H\) and her speed then reduces to \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). For the first 2 s her motion is modelled as that of a particle falling freely under gravity. For the next 5 s the model is motion with constant deceleration, so that her speed is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the end of this period. For the rest of the time before she reaches the ground, the model is motion with constant speed of \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Sketch a speed-time graph to illustrate her motion from \(H\) to the ground.
2. Find her speed when the parachute opens. A safety rule states that the helicopter must be high enough to allow the parachute to open and for the speed of a parachutist to reduce to \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) before reaching the ground. Using the assumptions made in the above model,
3. find the minimum height of \(H\) for which the woman can make a drop without breaking this safety rule. Given that \(H\) is 125 m above the ground when the woman starts her drop,
4. find the total time taken for her to reach the ground.
5. State one way in which the model could be refined to make it more realistic.
   (1 mark)

2. 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 \(a \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find

1. the value of \(a\),
2. 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.
3. Find the height of the rocket above the ground 5 s after it has left the ground.

3. A car moves along a horizontal straight road, passing two points \(A\) and \(B\). At \(A\) the speed of the car is \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). 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 \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). 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 \(V \mathrm {~m} \mathrm {~s} ^ { - 1 } ( V > 15 )\). He then maintains the car at a constant speed of \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Moving at this constant speed, the car passes \(B\) after a further 22 s .

1. Sketch, in the space below, a speed-time graph to illustrate the motion of the car as it moves from \(A\) to \(B\).
2. Find the time taken for the car to move from \(A\) to \(B\). The distance from \(A\) to \(B\) is 1 km .
3. Find the value of \(V\).

7. \begin{figure}[h] \end{figure} 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 \(\mu\). The particles are released from rest with the string taut. Immediately after release, the magnitude of the acceleration of \(A\) and \(B\) is \(\frac { 4 } { 9 } g\). By writing down separate equations of motion for \(A\) and \(B\),

1. find the tension in the string immediately after the particles begin to move,
2. show that \(\mu = \frac { 2 } { 3 }\). When \(B\) has fallen a distance \(h\), it hits the ground and does not rebound. Particle \(A\) is then a distance \(\frac { 1 } { 3 } h\) from \(P\).
3. Find the speed of \(A\) as it reaches \(P\).
4. State how you have used the information that the string is light.

2. A small ball is projected vertically upwards from ground level with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball takes 4 s to return to ground level.

1. Draw, in the space below, a velocity-time graph to represent the motion of the ball during the first 4 s .
2. The maximum height of the ball above the ground during the first 4 s is 19.6 m . Find the value of \(u\).

4 A particle moves in a straight line. Its velocity \(t \mathrm {~s}\) after leaving a fixed point on the line is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), where \(v = t + 0.1 t ^ { 2 }\). Find

1. an expression for the acceleration of the particle at time \(t\),
2. the distance travelled by the particle from time \(t = 0\) until the instant when its acceleration is \(2.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

6 \begin{figure}[h] \end{figure} A cyclist \(P\) travels along a straight road starting from rest at \(A\) and accelerating at \(2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) up to a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). He continues at a constant speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), passing through the point \(B 20 \mathrm {~s}\) after leaving \(A\). Fig. 1 shows the ( \(t , v\) ) graph of \(P\) 's journey for \(0 \leqslant t \leqslant 20\). Find

1. the time for which \(P\) is accelerating,
2. the distance \(A B\). \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{5b10afa1-1c45-4370-a0e6-ad8fd626df9a-4_607_937_1420_605} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} Another cyclist \(Q\) travels along the same straight road in the opposite direction. She starts at rest from \(B\) at the same instant that \(P\) leaves \(A\). Cyclist \(Q\) accelerates at \(2 \mathrm {~ms} ^ { - 2 }\) up to a speed of \(8 \mathrm {~ms} ^ { - 1 }\) and continues at a constant speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), passing through the point \(A 20 \mathrm {~s}\) after leaving \(B\). Fig. 2 shows the \(( t , x )\) graph of \(Q\) 's journey for \(0 \leqslant t \leqslant 20\), where \(x\) is the displacement of \(Q\) from \(A\) towards \(B\).
3. Sketch a copy of Fig. 1 and add to your copy a sketch of the ( \(t , v\) ) graph of \(Q\) 's journey for \(0 \leqslant t \leqslant 20\).
4. Sketch a copy of Fig. 2 and add to your copy a sketch of the \(( t , x )\) graph of \(P\) 's journey for \(0 \leqslant t \leqslant 20\).
5. Find the value \(t\) at the instant that \(P\) and \(Q\) pass each other. \includegraphics[max width=\textwidth, alt={}, center]{5b10afa1-1c45-4370-a0e6-ad8fd626df9a-5_447_739_269_703} The upper edge of a smooth plane inclined at \(70 ^ { \circ }\) to the horizontal is joined to an edge of a rough horizontal table. Particles \(A\) and \(B\), of masses 0.3 kg and 0.2 kg respectively, are attached to the ends of a light inextensible string. The string passes over a smooth pulley which is fixed at the top of the smooth inclined plane. Particle \(A\) is held in contact with the rough horizontal table and particle \(B\) is in contact with the smooth inclined plane with the string taut (see diagram). The coefficient of friction between \(A\) and the horizontal table is 0.4 . Particle \(A\) is released from rest and the system starts to move.

1. Find the acceleration of \(A\) and the tension in the string. The string breaks when the speed of the particles is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Assuming \(A\) does not reach the pulley, find the distance travelled by \(A\) after the string breaks.
3. Assuming \(B\) does not reach the ground before \(A\) stops, find the distance travelled by \(B\) from the time the string breaks to the time that \(A\) stops.

7 \includegraphics[max width=\textwidth, alt={}, center]{db77a63a-6ff8-4fe5-bdd0-15afb7eb4866-4_419_419_274_735} Particles \(A\) and \(B\) are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley. The particles are released from rest, with the string taut, and \(A\) and \(B\) at the same height above a horizontal floor (see diagram). In the subsequent motion, \(A\) descends with acceleration \(1.4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and strikes the floor 0.8 s after being released. It is given that \(B\) never reaches the pulley.

1. Calculate the distance \(A\) moves before it reaches the floor and the speed of \(A\) immediately before it strikes the floor.
2. Show that \(B\) rises a further 0.064 m after \(A\) strikes the floor, and calculate the total length of time during which \(B\) is rising.
3. Sketch the ( \(t , v\) ) graph for the motion of \(B\) from the instant it is released from rest until it reaches a position of instantaneous rest.
4. Before \(A\) strikes the floor the tension in the string is 5.88 N . Calculate the mass of \(A\) and the mass of \(B\).
5. The pulley has mass 0.5 kg , and is held in a fixed position by a light vertical chain. Calculate the tension in the chain
   1. immediately before \(A\) strikes the floor,
   2. immediately after \(A\) strikes the floor.