3.02d Constant acceleration: SUVAT formulae

1. \begin{figure}[h] \end{figure} A particle of mass 0.8 kg is attached to one end of a light elastic spring, of natural length 2 m and modulus of elasticity 20 N . The other end of the spring is attached to a fixed point \(O\) on a smooth plane which is inclined at an angle \(\alpha\) to the horizontal, where tan \(\alpha = \frac { 3 } { 4 }\). The particle is held on the plane at a point which is 1.6 m down a line of greatest slope of the plane from \(O\), as shown in Figure 1. The particle is then released from rest. Find the initial acceleration of the particle.
(Total 6 marks)

1 A particle \(P\) is projected vertically downwards from a fixed point \(O\) with initial speed \(4.2 \mathrm {~ms} ^ { - 1 }\), and takes 1.5 s to reach the ground. Calculate

1. the speed of \(P\) when it reaches the ground,
2. the height of \(O\) above the ground,
3. the speed of \(P\) when it is 5 m above the ground.

6 A swimmer \(C\) swims with velocity \(v \mathrm {~ms} ^ { - 1 }\) in a swimming pool. At time \(t \mathrm {~s}\) after starting, \(v = 0.006 t ^ { 2 } - 0.18 t + k\), where \(k\) is a constant. \(C\) swims from one end of the pool to the other in 28.4 s .

1. Find the acceleration of \(C\) in terms of \(t\).
2. Given that the minimum speed of \(C\) is \(0.65 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), show that \(k = 2\).
3. Express the distance travelled by \(C\) in terms of \(t\), and calculate the length of the pool.

3 A particle is projected vertically upwards with velocity \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point 2.5 m above the ground.

1. Calculate the speed of the particle when it strikes the ground.
2. Calculate the time after projection when the particle reaches the ground.
3. Sketch on separate diagrams
   1. the \(( t , v )\) graph,
   2. the \(( t , x )\) graph,
      representing the motion of the particle.

5 \includegraphics[max width=\textwidth, alt={}, center]{4c6c9323-8238-4ec2-94a1-6e8188a34521-03_538_917_918_614} \(X\) is a point on a smooth plane inclined at \(\theta ^ { \circ }\) to the horizontal. \(Y\) is a point directly above the line of greatest slope passing through \(X\), and \(X Y\) is horizontal. A particle \(P\) is projected from \(X\) with initial speed \(4.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) down the line of greatest slope, and simultaneously a particle \(Q\) is released from rest at \(Y\). \(P\) moves with acceleration \(4.9 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), and \(Q\) descends freely under gravity (see diagram). The two particles collide at the point on the plane directly below \(Y\) at time \(T\) s after being set in motion.

1. (a) Express in terms of \(T\) the distances travelled by the particles before the collision.
   (b) Calculate \(\theta\).
   (c) Using the answers to parts (a) and (b), show that \(T = \frac { 2 } { 3 }\).
2. Calculate the speeds of the particles immediately before they collide.

5 \includegraphics[max width=\textwidth, alt={}, center]{2b3457b6-1fe9-4e67-91d4-a8bc4a5b1709-3_394_789_251_639} The diagram shows the ( \(t , v\) ) graph of an athlete running in a straight line on a horizontal track in a 100 m race. He starts from rest and has constant acceleration until he reaches a speed of \(15 \mathrm {~ms} ^ { - 1 }\) when \(t = T\). He maintains this constant speed until he decelerates at a constant rate of \(1.75 \mathrm {~ms} ^ { - 2 }\) for the final 4 s of the race. He completes the race in 10 s .

1. Calculate \(T\). The athlete races against a robot which has a displacement from the starting line of \(\left( 3 t ^ { 2 } - 0.2 t ^ { 3 } \right) \mathrm { m }\), at time \(t \mathrm {~s}\) after the start of the race.
2. Show that the speed of the robot is \(15 \mathrm {~ms} ^ { - 1 }\) when \(t = 5\).
3. Find the value of \(t\) for which the decelerations of the robot and the athlete are equal.
4. Verify that the athlete and the robot reach the finish line simultaneously.

6 A particle \(P\) of mass 0.3 kg is projected upwards along a line of greatest slope from the foot of a plane inclined at \(30 ^ { \circ }\) to the horizontal. The initial speed of \(P\) is \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the coefficient of friction is 0.15 . The particle \(P\) comes to instantaneous rest before it reaches the top of the plane.

1. Calculate the distance \(P\) moves up the plane.
2. Find the time taken by \(P\) to return from its highest position on the plane to the foot of the plane.
3. Calculate the change in the momentum of \(P\) between the instant that \(P\) leaves the foot of the plane and the instant that \(P\) returns to the foot of the plane.

7 \includegraphics[max width=\textwidth, alt={}, center]{2b3457b6-1fe9-4e67-91d4-a8bc4a5b1709-4_369_508_246_781} Particles \(P\) and \(Q\), of masses \(m \mathrm {~kg}\) and 0.05 kg respectively, are attached to the ends of a light inextensible string which passes over a smooth pulley. \(Q\) is attached to a particle \(R\) of mass 0.45 kg by a light inextensible string. The strings are taut, and the portions of the strings not in contact with the pulley are vertical. \(P\) is in contact with a horizontal surface when the particles are released from rest (see diagram). The tension in the string \(Q R\) is 2.52 N during the descent of \(R\).

1. (a) Find the acceleration of \(R\) during its descent.
   (b) By considering the motion of \(Q\), calculate the tension in the string \(P Q\) during the descent of \(R\).
2. Find the value of \(m\). \(R\) strikes the surface 0.5 s after release and does not rebound. During their subsequent motion, \(P\) does not reach the pulley and \(Q\) does not reach the surface.
3. Calculate the greatest height of \(P\) above the surface.

3 \includegraphics[max width=\textwidth, alt={}, center]{f5085265-5258-45d4-8233-6bd68f8e9034-2_300_501_799_790} A particle \(P\) of mass 0.25 kg moves upwards with constant speed \(u \mathrm {~ms} ^ { - 1 }\) along a line of greatest slope on a smooth plane inclined at \(30 ^ { \circ }\) to the horizontal. The pulling force acting on \(P\) has magnitude \(T \mathrm {~N}\) and acts at an angle of \(20 ^ { \circ }\) to the line of greatest slope (see diagram). Calculate

1. the value of \(T\),
2. the magnitude of the contact force exerted on \(P\) by the plane. The pulling force \(T \mathrm {~N}\) acting on \(P\) is suddenly removed, and \(P\) comes to instantaneous rest 0.4 s later.
3. Calculate \(u\).

5 \includegraphics[max width=\textwidth, alt={}, center]{f5085265-5258-45d4-8233-6bd68f8e9034-3_462_405_258_845} A small smooth pulley is suspended from a fixed point by a light chain. A light inextensible string passes over the pulley. Particles \(P\) and \(Q\), of masses 0.3 kg and \(m \mathrm {~kg}\) respectively, are attached to the opposite ends of the string. The particles are released from rest at a height of 0.2 m above horizontal ground with the string taut; the portions of the string not in contact with the pulley are vertical (see diagram). \(P\) strikes the ground with speed \(1.4 \mathrm {~ms} ^ { - 1 }\). Subsequently \(P\) remains on the ground, and \(Q\) does not reach the pulley.

1. Calculate the acceleration of \(P\) while it is in motion and the corresponding tension in the string.
2. Find the value of \(m\).
3. Calculate the greatest height of \(Q\) above the ground.
4. It is given that the mass of the pulley is 0.5 kg . State the magnitude of the tension in the chain which supports the pulley
   1. when \(P\) is in motion,
   2. when \(P\) is at rest on the ground and \(Q\) is moving upwards.

2 Particles \(P\) and \(Q\), of masses 0.45 kg and \(m \mathrm {~kg}\) respectively, are attached to the ends of a light inextensible string which passes over a small smooth pulley. The particles are released from rest with the string taut and both particles 0.36 m above a horizontal surface. \(Q\) descends with acceleration \(0.98 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). When \(Q\) strikes the surface, it remains at rest.

1. Calculate the tension in the string while both particles are in motion.
2. Find the value of \(m\).
3. Calculate the speed at which \(Q\) strikes the surface.
4. Calculate the greatest height of \(P\) above the surface. (You may assume that \(P\) does not reach the pulley.)

4 \includegraphics[max width=\textwidth, alt={}, center]{ce4c43e6-da4f-4c02-ab0f-01a21717949c-2_657_1495_1539_324} A car travelling on a straight road accelerates from rest to a speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 6 s . It continues at constant speed for 11 s and then decelerates to rest in 2 s . The driver gets out of the car and walks at a speed of \(2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) for 20 s back to a shop which he enters. Some time later he leaves the shop and jogs to the car at a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). He arrives at the vehicle 60 s after it began to accelerate from rest. The diagram, which has six straight line segments, shows the \(( t , v )\) graph for the motion of the driver.

1. Calculate the initial acceleration and final deceleration of the car.
2. Calculate the distance the car travels.
3. Calculate the length of time the driver is in the shop.

5 \includegraphics[max width=\textwidth, alt={}, center]{ce4c43e6-da4f-4c02-ab0f-01a21717949c-3_362_1065_258_539} Three particles \(P , Q\) and \(R\) lie on a line of greatest slope of a smooth inclined plane. \(P\) has mass 0.5 kg and initially is at the foot of the plane. \(R\) has mass 0.3 kg and initially is at the top of the plane. \(Q\) has mass 0.2 kg and is between \(P\) and \(R\) (see diagram). \(P\) is projected up the line of greatest slope with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at the instant when \(Q\) and \(R\) are released from rest. Each particle has an acceleration of \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) down the plane.

1. \(P\) and \(Q\) collide 0.4 s after being set in motion. Immediately after the collision \(Q\) moves up the plane with speed \(3.2 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the speed and direction of motion of \(P\) immediately after the collision.
2. 0.6 s after its collision with \(P , Q\) collides with \(R\) and the two particles coalesce. Find the speed and direction of motion of the combined particle immediately after the collision

2 A particle is projected vertically upwards with speed \(7 \mathrm {~ms} ^ { - 1 }\) from a point on the ground.

1. Find the speed of the particle and its distance above the ground 0.4 s after projection.
2. Find the total distance travelled by the particle in the first 0.9 s after projection.

4 A block \(B\) of weight 28 N is pulled at constant speed across a rough horizontal surface by a force of magnitude 14 N inclined at \(30 ^ { \circ }\) above the horizontal.

1. Show that the coefficient of friction between the block and the surface is 0.577 , correct to 3 significant figures. The 14 N force is suddenly removed, and the block decelerates, coming to rest after travelling a further 3.2 m .
2. Calculate the speed of the block at the instant the 14 N force was removed.

5 \includegraphics[max width=\textwidth, alt={}, center]{f0813713-d677-4ed7-87e1-971a64bdb6ff-3_291_182_799_945} Particles \(P\) and \(Q\), of masses 0.4 kg and \(m \mathrm {~kg}\) respectively, are joined by a light inextensible string which passes over a smooth pulley. The particles are released from rest at the same height above a horizontal surface; the string is taut and the portions of the string not in contact with the pulley are vertical (see diagram). \(Q\) begins to descend with acceleration \(2.45 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) and reaches the surface 0.3 s after being released. Subsequently, \(Q\) remains at rest and \(P\) never reaches the pulley.

1. Calculate the tension in the string while \(Q\) is in motion.
2. Calculate the momentum lost by \(Q\) when it reaches the surface.
3. Calculate the greatest height of \(P\) above the surface. \section*{[Questions 6 and 7 are printed overleaf.]}

2 A particle \(P\) is projected vertically upwards and reaches its greatest height 0.5 s after the instant of projection. Calculate

1. the speed of projection of \(P\),
2. the greatest height of \(P\) above the point of projection. It is given that the point of projection is 0.539 m above the ground.
3. Find the speed of \(P\) immediately before it strikes the ground.

4 \includegraphics[max width=\textwidth, alt={}, center]{b7f05d10-9d3c-4098-846d-ca6511c75c5d-3_298_540_262_735} The diagram shows the \(( t , v )\) graph of a car moving along a straight road, where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the velocity of the car at time \(t \mathrm {~s}\) after it passes through the point \(A\). The car passes through \(A\) with velocity \(18 \mathrm {~ms} ^ { - 1 }\), and moves with constant acceleration \(2.4 \mathrm {~ms} ^ { - 2 }\) until \(t = 5\). The car subsequently moves with constant velocity until it is 300 m from \(A\). When the car is more than 300 m from \(A\), it has constant deceleration \(6 \mathrm {~ms} ^ { - 2 }\), until it comes to rest.

1. Find the greatest speed of the car.
2. Calculate the value of \(t\) for the instant when the car begins to decelerate.
3. Calculate the distance from \(A\) of the car when it is at rest.

5 A particle \(P\) is projected with speed \(u \mathrm {~ms} ^ { - 1 }\) from the top of a smooth inclined plane of length \(2 d\) metres. After its projection \(P\) moves downwards along a line of greatest slope with acceleration \(4 \mathrm {~ms} ^ { - 2 }\). At the instant 3 s after projection \(P\) has moved half way down the plane. \(P\) reaches the foot of the plane 5 s after the instant of projection.

1. Form two simultaneous equations in \(u\) and \(d\), and hence calculate the speed of projection of \(P\) and the length of the plane.
2. Find the inclination of the plane to the horizontal.
3. Given that the contact force exerted on \(P\) by the plane has magnitude 6 N , calculate the mass of \(P\).

1 A particle \(P\) is projected vertically downwards with speed \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point 30 m above the ground.

1. Calculate the speed of \(P\) when it reaches the ground.
2. Find the distance travelled by \(P\) in the first 0.4 s of its motion.
3. Calculate the time taken for \(P\) to travel the final 15 m of its descent.

1 A particle is travelling in a straight line. Its velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t\) seconds is given by $$v = 6 + 4 t \quad \text { for } 0 \leqslant t \leqslant 5$$

1. Write down the initial velocity of the particle and find the acceleration for \(0 \leqslant t \leqslant 5\).
2. Write down the velocity of the particle when \(t = 5\). Find the distance travelled in the first 5 seconds. For \(5 \leqslant t \leqslant 15\), the acceleration of the particle is \(3 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
3. Find the total distance travelled by the particle during the 15 seconds.

6 Small stones A and B are initially in the positions shown in Fig. 6 with B a height \(H \mathrm {~m}\) directly above A. \begin{figure}[h] \end{figure} At the instant when B is released from rest, A is projected vertically upwards with a speed of \(29.4 \mathrm {~m} \mathrm {~s} \mathrm {~s} ^ { - 1 }\). Air resistance may be neglected. The stones collide \(T\) seconds after they begin to move. At this instant they have the same speed, \(V \mathrm {~m} \mathrm {~s} ^ { - 1 }\), and A is still rising. By considering when the speed of A upwards is the same as the speed of B downwards, or otherwise, show that \(T = 1.5\) and find the values of \(V\) and \(H\). Section B (36 marks)

1 An egg falls from rest a distance of 75 cm to the floor.
Neglecting air resistance, at what speed does it hit the floor?

6 A small ball is kicked off the edge of a jetty over a calm sea. Air resistance is negligible. Fig. 6 shows

• the point of projection, O,
• the initial horizontal and vertical components of velocity,
• the point A on the jetty vertically below O and at sea level,
• the height, OA, of the jetty above the sea.

\begin{figure}[h] \end{figure} The time elapsed after the ball is kicked is \(t\) seconds.

1. Find an expression in terms of \(t\) for the height of the ball above O at time \(t\). Find also an expression for the horizontal distance of the ball from O at this time.
2. Determine how far the ball lands from A .

1 A pellet is fired vertically upwards at a speed of \(11 \mathrm {~ms} ^ { - 1 }\). Assuming that air resistance may be neglected, calculate the speed at which the pellet hits a ceiling 2.4 m above its point of projection.