3.02d Constant acceleration: SUVAT formulae

3 Two small spheres \(P\) and \(Q\) have masses 0.1 kg and 0.2 kg respectively. The spheres are moving directly towards each other on a horizontal plane and collide. Immediately before the collision \(P\) has speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(Q\) has speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Immediately after the collision the spheres move away from each other, \(P\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(Q\) with speed \(( 3.5 - u ) \mathrm { m } \mathrm { s } ^ { - 1 }\).

1. Find the value of \(u\). After the collision the spheres both move with deceleration of magnitude \(5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) until they come to rest on the plane.
2. Find the distance \(P Q\) when both \(P\) and \(Q\) are at rest.

4 A particle moves downwards on a smooth plane inclined at an angle \(\alpha\) to the horizontal. The particle passes through the point \(P\) with speed \(u \mathrm {~ms} ^ { - 1 }\). The particle travels 2 m during the first 0.8 s after passing through \(P\), then a further 6 m in the next 1.2 s . Find

1. the value of \(u\) and the acceleration of the particle,
2. the value of \(\alpha\) in degrees.

1 A block \(B\) of mass 3 kg moves with deceleration \(1.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) in a straight line on a rough horizontal surface. The initial speed of \(B\) is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Calculate

1. the time for which \(B\) is in motion,
2. the distance travelled by \(B\) before it comes to rest,
3. the coefficient of friction between \(B\) and the surface.

6 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 \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car travels 14 m in the 2 seconds after passing the sign; 5 seconds after passing the sign it has a speed of \(19 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

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? Section B (36 marks)

7 A horizontal force of 24 N acts on a block of mass 12 kg on a horizontal plane. The block is initially at rest. This situation is first modelled assuming the plane is smooth.

1. Write down the acceleration of the block according to this model. The situation is now modelled assuming a constant resistance to motion of 15 N .
2. Calculate the acceleration of the block according to this new model. How much less distance does the new model predict that the block will travel in the first 4 seconds? The 24 N force is removed and the block slides down a slope at \(5 ^ { \circ }\) to the horizontal. The speed of the block at the top of the slope is \(1.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), as shown in Fig. 7. The answers to parts (iii) and (iv) should be found using the assumption that the resistance to the motion of the block is still a constant 15 N . \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{52d6c914-b204-4587-a82e-fbab6693fcf8-5_255_901_1128_575} \captionsetup{labelformat=empty} \caption{Fig. 7}
   \end{figure}
3. Calculate the acceleration of the block in the direction of its motion.
4. For how much time does the block slide down the slope before coming to rest and how far does it slide in that time? Measurements show that the block actually comes to rest in 3.5 seconds.
5. Assuming that the error in the prediction is due only to the value of the resistance, calculate the true value of the resistance.

3 Two cars, P and Q, are being crashed as part of a film 'stunt'.
At the start

• P is travelling directly towards Q with a speed of \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\),
• Q is instantaneously at rest and has an acceleration of \(4 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) directly towards P . \(P\) continues with the same velocity and \(Q\) continues with the same acceleration. The cars collide \(T\) seconds after the start.
  1. Find expressions in terms of \(T\) for how far each of the cars has travelled since the start.

At the start, \(P\) is 90 m from \(Q\).

Show that \(T ^ { 2 } + 4 T - 45 = 0\) and hence find \(T\).

6 Robin is driving a car of mass 800 kg along a straight horizontal road at a speed of \(40 \mathrm {~ms} ^ { - 1 }\).
Robin applies the brakes and the car decelerates uniformly; it comes to rest after travelling a distance of 125 m .

1. Show that the resistance force on the car when the brakes are applied is 5120 N .
2. Find the time the car takes to come to rest. For the rest of this question, assume that when Robin applies the brakes there is a constant resistance force of 5120 N on the car. The car returns to its speed of \(40 \mathrm {~ms} ^ { - 1 }\) and the road remains straight and horizontal.
   Robin sees a red light 155 m ahead, takes a short time to react and then applies the brakes.
   The car comes to rest before it reaches the red light.
3. Show that Robin's reaction time is less than 0.75 s . The 'stopping distance' is the total distance travelled while a driver reacts and then applies the brakes to bring the car to rest. For the rest of this question, assume that Robin is still driving the car described above and has a reaction time of 0.675 s . (This is the figure used in calculating the stopping distances given in the Highway Code.)
4. Calculate the stopping distance when Robin is driving at \(20 \mathrm {~ms} ^ { - 1 }\) on a horizontal road. The car then travels down a hill which has a slope of \(5 ^ { \circ }\) to the horizontal.
5. Find the stopping distance when Robin is driving at \(20 \mathrm {~ms} ^ { - 1 }\) down this hill.
6. By what percentage is the stopping distance increased by the fact that the car is going down the hill? Give your answer to the nearest \(1 \%\).

3 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.

4 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 \(9 \mathrm {~ms} ^ { - 1 }\) 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 .

2 A car passes a point A travelling at \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Its motion over the next 45 seconds is modelled as follows.

• The car's speed increases uniformly from \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over the first 10 s .
• Its speed then increases uniformly to \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) over the next 15 s .
• The car then maintains this speed for a further 20 s at which time it reaches the point B .
  1. Sketch a speed-time graph to represent this motion.
  2. Calculate the distance from A to B .
  3. When it reaches the point B , the car is brought uniformly to rest in \(T\) seconds. The total distance from A is now 1700 m . Calculate the value of \(T\).

4 \begin{figure}[h] \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?

4 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 .

2 \begin{figure}[h] \end{figure} A particle moves on the straight line shown in Fig. 2. The positive direction is indicated on the diagram. The time, \(t\), is measured in seconds. The particle has constant acceleration, \(a \mathrm {~ms} ^ { - 2 }\). Initially it is at the point O and has velocity \(u \mathrm {~ms} ^ { - 1 }\).
When \(t = 2\), the particle is at A where OA is 12 m . The particle is also at A when \(t = 6\).

1. Write down two equations in \(u\) and \(a\) and solve them.
2. The particle changes direction when it is at B . Find the distance AB .

4 Fig. 4 illustrates points \(A , B\) and \(C\) on a straight race track. The distance \(A B\) is 300 m and \(A C\) 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 20 s 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
An egg falls from rest a distance of 75 cm to the floor.
Neglecting air resistance, at what speed does it hit the floor?

1. Particle \(P\) has mass \(m\) and particle \(Q\) has mass \(5 m\).

The particles are moving in the same direction along the same straight line on a smooth horizontal surface. Particle \(P\) collides directly with particle \(Q\).
Immediately before the collision, the speed of \(P\) is \(6 u\) and the speed of \(Q\) is \(u\).
Immediately after the collision, the speed of \(P\) is \(x\) and the speed of \(Q\) is \(y\).
The direction of motion of \(P\) is reversed as a result of the collision.
The coefficient of restitution between \(P\) and \(Q\) is \(e\).

1. Find the complete range of possible values of \(e\). Given that \(e = \frac { 3 } { 5 }\)
2. find the total kinetic energy lost in the collision between \(P\) and \(Q\). After the collision, \(Q\) hits a smooth fixed vertical wall that is perpendicular to the direction of motion of \(Q\). Particle \(Q\) rebounds.
   The coefficient of restitution between \(Q\) and the wall is \(f\).
   Given that there is a second collision between \(P\) and \(Q\),
3. find the complete range of possible values of \(f\).

A particle \(P\) moves on the \(x\)-axis. The acceleration of \(P\), in the positive \(x\) direction at time \(t\) seconds, is \(( 2 t - 3 ) \mathrm { m } \mathrm { s } ^ { - 2 }\). The velocity of \(P\), in the positive \(x\) direction at time \(t\) seconds, is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). When \(t = 0 , v = 2\)

1. Find \(v\) in terms of \(t\). The particle is instantaneously at rest at times \(t _ { 1 }\) seconds and \(t _ { 2 }\) seconds, where \(t _ { 1 } < t _ { 2 }\).
2. Find the values \(t _ { 1 }\) and \(t _ { 2 }\).
3. Find the distance travelled by \(P\) between \(t = t _ { 1 }\) and \(t = t _ { 2 }\).
   

7. A particle \(P\) is projected from a fixed point \(A\) with speed \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) above the horizontal and moves freely under gravity. When \(P\) passes through the point \(B\) on its path, it has speed \(7 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. By considering energy, find the vertical distance between \(A\) and \(B\). The minimum speed of \(P\) on its path from \(A\) to \(B\) is \(2.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the size of angle \(\alpha\).
3. Find the horizontal distance between \(A\) and \(B\).

6. [In this question the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a vertical plane, \(\mathbf { i }\) being horizontal and \(\mathbf { j }\) being vertically upwards.] At \(t = 0\) a particle \(P\) is projected from a fixed point \(O\) with velocity ( \(7 \mathbf { i } + 7 \sqrt { 3 } \mathbf { j }\) ) \(\mathrm { m } \mathrm { s } ^ { - 1 }\). The particle moves freely under gravity. The position vector of a point on the path of \(P\) is \(( x \mathbf { i } + y \mathbf { j } ) \mathrm { m }\) relative to \(O\).

1. Show that $$y = \sqrt { 3 } x - \frac { g } { 98 } x ^ { 2 }$$
2. Find the direction of motion of \(P\) when it passes through the point on the path where \(x = 20\) At time \(T\) seconds \(P\) passes through the point with position vector \(( 2 \lambda \mathbf { i } + \lambda \mathbf { j } ) \mathrm { m }\) where \(\lambda\) is a positive constant.
3. Find the value of \(T\).
   DO NOT WIRITE IN THIS AREA

1. \hspace{0pt} [In this question, the unit vectors \(\mathbf { i }\) and \(\mathbf { j }\) are in a vertical plane, \(\mathbf { i }\) being horizontal and j being vertically upwards.]

\begin{figure}[h] \end{figure} A golf ball is hit from a point \(O\) on horizontal ground and is modelled as a particle moving freely under gravity. The initial velocity of the ball is \(( 2 u \mathbf { i } + u \mathbf { j } ) \mathrm { ms } ^ { - 1 }\) The ball first hits the horizontal ground at a point which is 80 m from \(O\), as shown in Figure 3. Use the model to

1. show that \(u = 14\)
2. find the total time, while the ball is in the air, for which the speed of the ball is greater than \(7 \sqrt { 17 } \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

4. At time \(t = 0\) a ball is projected from a fixed point \(A\) on horizontal ground to hit a target. The ball is projected from \(A\) with speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\theta ^ { \circ }\) to the horizontal. At time \(t = 2 \mathrm {~s}\) the ball hits the target. At the instant when it hits the target, the ball is travelling downwards at \(30 ^ { \circ }\) below the horizontal with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The ball is modelled as a particle moving freely under gravity and the target is modelled as the point \(T\).

1. Find
   1. the value of \(\theta\),
   2. the value of \(u\). The height of \(T\) above the ground is \(h\) metres.
2. Find the value of \(h\).
3. Find the length of time for which the ball is more than \(h\) metres above the ground during the flight from \(A\) to \(T\).

7. \begin{figure}[h] \end{figure} A ball is thrown from a point \(A\) at a target, which is on horizontal ground. The point \(A\) is 12 m above the point \(O\) on the ground. The ball is thrown from \(A\) with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(30 ^ { \circ }\) below the horizontal. The ball is modelled as a particle and the target as a point \(T\). The distance \(O T\) is 15 m . The ball misses the target and hits the ground at the point \(B\), where \(O T B\) is a straight line, as shown in Figure 4. Find

1. the time taken by the ball to travel from \(A\) to \(B\),
2. the distance \(T B\). The point \(X\) is on the path of the ball vertically above \(T\).
3. Find the speed of the ball at \(X\).

6. \begin{figure}[h] \end{figure} A child playing cricket on horizontal ground hits the ball towards a fence 10 m away. The ball moves in a vertical plane which is perpendicular to the fence. The ball just passes over the top of the fence, which is 2 m above the ground, as shown in Figure 3. The ball is modelled as a particle projected with initial speed \(u \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from point \(O\) on the ground at an angle \(\alpha\) to the ground.

1. By writing down expressions for the horizontal and vertical distances, from \(O\) of the ball \(t\) seconds after it was hit, show that $$2 = 10 \tan \alpha - \frac { 50 g } { u ^ { 2 } \cos ^ { 2 } \alpha }$$ Given that \(\alpha = 45 ^ { \circ }\),
2. find the speed of the ball as it passes over the fence.

7. \begin{figure}[h] \end{figure} A small stone is projected from a point \(O\) at the top of a vertical cliff \(O A\). The point \(O\) is 52.5 m above the sea. The stone rises to a maximum height of 10 m above the level of \(O\) before hitting the sea at the point \(B\), where \(A B = 50 \mathrm {~m}\), as shown in Figure 4. The stone is modelled as a particle moving freely under gravity.

1. Show that the vertical component of the velocity of projection of the stone is \(14 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Find the speed of projection.
3. Find the time after projection when the stone is moving parallel to \(O B\).

2. \begin{figure}[h] \end{figure} A light elastic spring has natural length \(l\) and modulus of elasticity \(\lambda\) One end of the spring is attached to a point \(A\) on a smooth plane.
The plane is inclined at angle \(\theta\) to the horizontal, where \(\tan \theta = \frac { 5 } { 12 }\) A particle \(P\) of mass \(m\) is attached to the other end of the spring. Initially \(P\) is held at the point \(B\) on the plane, where \(A B\) is a line of greatest slope of the plane. The point \(B\) is lower than \(A\) and \(A B = 2 l\), as shown in Figure 1 .
The particle is released from rest at \(B\) and first comes to instantaneous rest at the point \(C\) on \(A B\), where \(A C = 0.7 l\)

1. Use the principle of conservation of mechanical energy to show that $$\lambda = \frac { 100 } { 91 } m g$$
2. Find the acceleration of \(P\) when it is released from rest at \(B\).