3.02d Constant acceleration: SUVAT formulae

2 The speed of a 100 metre runner in \(\mathrm { m } \mathrm { s } ^ { - 1 }\) is measured electronically every 4 seconds.
The measurements are plotted as points on the speed-time graph in Fig. 6. The vertical dotted line is drawn through the runner's finishing time. Fig. 6 also illustrates Model P in which the points are joined by straight lines. \begin{figure}[h] \end{figure}

1. Use Model P to estimate
   (A) the distance the runner has gone at the end of 12 seconds,
   (B) how long the runner took to complete 100 m . A mathematician proposes Model Q in which the runner's speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at time \(t \mathrm {~s}\), is given by $$v = \frac { 5 } { 2 } t - \frac { 1 } { 8 } t ^ { 2 } .$$
2. Verify that Model Q gives the correct speed for \(t = 8\).
3. Use Model Q to estimate the distance the runner has gone at the end of 12 seconds.
4. The runner was timed at 11.35 seconds for the 100 m . Which model places the runner closer to the finishing line at this time? In this question take \(g\) as \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).
   A small ball is released from rest. It falls for 2 seconds and is then brought to rest over the next 5 seconds. This motion is modelled in the speed-time graph Fig. 6. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{4f80ea36-001f-4a00-849f-542f5072516b-3_658_1101_281_503} \captionsetup{labelformat=empty} \caption{Fig. 6}
   \end{figure} For this model,
5. calculate the distance fallen from \(t = 0\) to \(t = 7\),
6. find the acceleration of the ball from \(t = 2\) to \(t = 6\), specifying the direction,
7. obtain an expression in terms of \(t\) for the downward speed of the ball from \(t = 2\) to \(t = 6\),
8. state the assumption that has been made about the resistance to motion from \(t = 0\) to \(t = 2\). The part of the motion from \(t = 2\) to \(t = 7\) is now modelled by \(v = - \frac { 3 } { 2 } t ^ { 2 } + \frac { 19 } { 2 } t + 7\).
9. Verify that \(v\) agrees with the values given in Fig, 6 at \(t = 2 , t = 6\) and \(t = 7\).
10. Calculate the distance fallen from \(t = 2\) to \(t = 7\) according to this model.

2 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 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 \(3 \leqslant t \leqslant 4\).

6 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 \(10 \mathrm {~ms} { } ^ { 1 }\) to \(30 \mathrm {~ms} { } ^ { 1 }\) 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 \(T\).

3 A cyclist starts from rest and takes 10 seconds to accelerate at a constant rate up to a speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). After travelling at this speed for 20 seconds, the cyclist then decelerates to rest at a constant rate over the next 5 seconds.

1. Sketch a velocity-time graph for the motion.
2. Calculate the distance travelled by the cyclist.

5 Fig. 8.1 shows a sledge of mass 40 kg . It is being pulled across a horizontal surface of deep snow by a light horizontal rope. There is a constant resistance to its motion. The tension in the rope is 120 N . \begin{figure}[h] \end{figure} The sledge is initially at rest. After 10 seconds its speed is \(5 \mathrm {~ms} ^ { - 1 }\).

1. Show that the resistance to motion is 100 N . When the speed of the sledge is \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the rope breaks.
   The resistance to motion remains 100 N .
2. Find the speed of the sledge
   (A) 1.6 seconds after the rope breaks,
   (B) 6 seconds after the rope breaks. The sledge is then pushed to the bottom of a ski slope. This is a plane at an angle of \(15 ^ { \circ }\) to the horizontal. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{9bff41e0-7be0-4932-ae50-a612abb3fe19-5_263_854_1391_637} \captionsetup{labelformat=empty} \caption{Fig. 8.2}
   \end{figure} The sledge is attached by a light rope to a winch at the top of the slope. The rope is parallel to the slope and has a constant tension of 200 N . Fig. 8.2 shows the situation when the sledge is part of the way up the slope. The ski slope is smooth.
3. Show that when the sledge has moved from being at rest at the bottom of the slope to the point when its speed is \(8 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), it has travelled a distance of 13.0 m (to 3 significant figures). When the speed of the sledge is \(8 \mathrm {~ms} ^ { - 1 }\), this rope also breaks.
4. Find the time between the rope breaking and the sledge reaching the bottom of the slope.

1 A golf ball is hit at an angle of \(60 ^ { \circ }\) to the horizontal from a point, O, on level horizontal ground. Its initial speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The standard projectile model, in which air resistance is neglected, is used to describe the subsequent motion of the golf ball. At time \(t \mathrm {~s}\) the horizontal and vertical components of its displacement from O are denoted by \(x \mathrm {~m}\) and \(y \mathrm {~m}\).

1. Write down equations for \(x\) and \(y\) in terms of \(t\).
2. Hence show that the equation of the trajectory is $$y = \sqrt { 3 } x - 0.049 x ^ { 2 }$$
3. Find the range of the golf ball.
4. A bird is hovering at position \(( 20,16 )\). Find whether the golf ball passes above it, passes below it or hits it.

2 A football is kicked with speed \(31 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(20 ^ { \circ }\) to the horizontal. It travels towards the goal which is 50 m away. The height of the crossbar of the goal is 2.44 m .

1. Does the ball go over the top of the crossbar? Justify your answer.
2. State one assumption that you made in answering part (i).

4 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.

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 .

7 Fig. 4 shows a particle projected over horizontal ground from a point O at ground level. The particle initially has a speed of \(32 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle \(\alpha\) to the horizontal. The particle is a horizontal distance of 44.8 m from O after 5 seconds. Air resistance should be neglected. \begin{figure}[h] \end{figure}

1. Write down an expression, in terms of \(\alpha\) and \(t\), for the horizontal distance of the particle from O at time \(t\) seconds after it is projected.
2. Show that \(\cos \alpha = 0.28\).
3. Calculate the greatest height reached by the particle.

2 A ball is kicked from ground level over horizontal ground. It leaves the ground at a speed of 25 ms 1 and at an angle \(\theta\) to the horizontal such that \(\cos \theta = 0.96\) and \(\sin \theta = 0.28\).

1. Show that the height, \(y \mathrm {~m}\), of the ball above the ground \(t\) seconds after projection is given by \(y = 7 t - 4.9 t ^ { 2 }\). Show also that the horizontal distance, \(x \mathrm {~m}\), travelled by this time is given by \(x = 24 t\).
2. Calculate the maximum height reached by the ball.
3. Calculate the times at which the ball is at half its maximum height. Find the horizontal distance travelled by the ball between these times.
4. Determine the following when \(t = 1.25\).
   (A) The vertical component of the velocity of the ball.
   (B) Whether the ball is rising or falling. (You should give a reason for your answer.)
   (C) The speed of the ball.
5. Show that the equation of the trajectory of the ball is $$y = \frac { 0.7 x } { 576 } ( 240 - 7 x )$$ Hence, or otherwise, find the range of the ball.

3 A particle is thrown vertically upwards and returns to its point of projection after 6 seconds. Air resistance is negligible. Calculate the speed of projection of the particle and also the maximum height it reaches.

4 You should neglect air resistance in this question.
A small stone is projected from ground level. The maximum height of the stone above horizontal ground is 22.5 m .

1. Show that the vertical component of the initial velocity of the stone is \(21 \mathrm {~ms} { } ^ { 1 }\). The speed of projection is \(28 \mathrm {~ms} { } ^ { 1 }\).
2. Find the angle of projection of the stone.
3. Find the horizontal range of the stone.

5 In this question take the value of \(\boldsymbol { g }\) to be \(\mathbf { 1 0 ~ } \mathbf { m ~ s } ^ { \mathbf { 2 } }\). \(\Lambda\) particle \(\Lambda\) is projected over horizontal ground from a point P which is 9 m above a point O on the ground. The initial velocity has horizontal and vertical components of \(10 \mathrm {~ms} ^ { - 1 }\) and \(12 \mathrm {~ms} ^ { - 1 }\) respectively, as shown in Fig. 7. The trajectory of the particle meets the ground at X. Air resistance may be neglected. \begin{figure}[h] \end{figure}

1. Calculate the specd of projection \(u \mathrm {~ms} ^ { - 1 }\) and the angle of projection \(\theta ^ { \circ }\).
2. Show that, \(t\) seconds after projection, the height of particle A above the ground is \(9 + 12 t - 5 t ^ { 2 }\). Write down an expression in terms of \(t\) for the horizontal distance of the particle from O at this time.
3. Calculate the maximum height of particle \(\Lambda\) above the point of projection.
4. Calculate the distance OX . \(\wedge\) second particle, \(B\), is projected from \(O\) with speed \(20 \mathrm {~ms} ^ { - 1 }\) at \(60 ^ { \circ }\) to the horizontal. The trajectories of A and B are in the same vertical plane. Particles A and B are projected at the same time.
5. Show that the horizontal displacements of A and B are always cqual.
6. Show that, \(t\) seconds after projection, the height of particle B above the ground is \(10 \sqrt { 3 } t - 5 t ^ { 2 }\).
7. Show that the particles collide 1.7 seconds after projection (correct to two significant figures).

6 Ali is throwing flat stones onto water, hoping that they will bounce, as illustrated in Fig. 5.
Ali throws one stone from a height of 1.225 m above the water with initial speed \(20 \mathrm {~ms} ^ { - 1 }\) in a horizontal direction. Air resistance should be neglected. \begin{figure}[h] \end{figure}

1. Find the time it takes for the stone to reach the water.
2. Find the speed of the stone when it reaches the water and the angle its trajectory makes with the horizontal at this time.

1 A small firework is fired from a point O at ground level over horizontal ground. The highest point reached by the firework is a horizontal distance of 60 m from O and a vertical distance of 40 m from O , as shown in Fig. 7. Air resistance is negligible. \begin{figure}[h] \end{figure} The initial horizontal component of the velocity of the firework is \(21 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Calculate the time for the firework to reach its highest point and show that the initial vertical component of its velocity is \(28 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
2. Show that the firework is \(\left( 28 t - 4.9 t ^ { 2 } \right) \mathrm { m }\) above the ground \(t\) seconds after its projection. When the firework is at its highest point it explodes into several parts. Two of the parts initially continue to travel horizontally in the original direction, one with the original horizontal speed of \(21 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and the other with a quarter of this speed.
3. State why the two parts are always at the same height as one another above the ground and hence find an expression in terms of \(t\) for the distance between the parts \(t\) seconds after the explosion.
4. Find the distance between these parts of the firework
   (A) when they reach the ground,
   (B) when they are 10 m above the ground.
5. Show that the cartesian equation of the trajectory of the firework before it explodes is \(y = \frac { 1 } { 90 } \left( 120 x - x ^ { 2 } \right)\), referred to the coordinate axes shown in Fig. 7.

1 A girl throws a small stone with initial speed \(14 \mathrm {~ms} { } ^ { 1 }\) at an angle of \(60 ^ { \circ }\) to the horizontal from a point 1 m above the ground. She throws the stone directly towards a vertical wall of height 6 m standing on horizontal ground. The point O is on the ground directly below the point of projection, as shown in Fig. 8. Air resistance is negligible. \begin{figure}[h] \end{figure}

1. Write down an expression in terms of \(t\) for the horizontal displacement of the stone from O , \(t\) seconds after projection. Find also an expression for the height of the stone above O at this time. The stone is at the top of its trajectory when it passes over the wall.
2. (A) Find the time it takes for the stone to reach its highest point.
   (B) Calculate the distance of O from the base of the wall.
   (C) Show that the stone passes over the wall with 2.5 m clearance.
3. Find the cartesian equation of the trajectory of the stone referred to the horizontal and vertical axes, \(\mathrm { O } x\) and \(\mathrm { O } y\). There is no need to simplify your answer. The girl now moves away a further distance \(d \mathrm {~m}\) from the wall. She throws a stone as before and it just passes over the wall.
4. Calculate \(d\).

2 A particle is projected vertically upwards from a point O at \(21 \mathrm {~ms} ^ { - 1 }\).

1. Calculate the greatest height reached by the particle. When this particle is at its highest point, a second particle is projected vertically upwards from \(O\) at \(15 \mathrm {~ms} ^ { - 1 }\).
2. Show that the particles collide 1.5 seconds later and determine the height above O at which the collision takes place.

3 The trajectory ABCD of a small stone moving with negligible air resistance is shown in Fig. 7. AD is horizontal and BC is parallel to AD . The stone is projected from A with speed \(40 \mathrm {~ms} ^ { - 1 }\) at \(50 ^ { \circ }\) to the horizontal. \begin{figure}[h] \end{figure}

1. Write down an expression for the horizontal displacement from A of the stone \(t\) seconds after projection. Write down also an expression for the vertical displacement at time \(t\).
2. Show that the stone takes 6.253 seconds (to three decimal places) to travel from A to D . Calculate the range of the stone. You are given that \(X = 30\).
3. Calculate the time it takes the stone to reach B . Hence determine the time for it to travel from A to C.
4. Calculate the direction of the motion of the stone at C .

1 A football is kicked from horizontal ground with speed \(20 \mathrm {~ms} ^ { - 1 }\) at an angle of \(\theta ^ { \circ }\) above the horizontal. The greatest height the football reaches above ground level is 2.44 m . By modelling the football as a particle and ignoring air resistance, find

1. the value of \(\theta\),
2. the range of the football.

8 A child is trying to throw a small stone to hit a target painted on a vertical wall. The child and the wall are on horizontal ground. The child is standing a horizontal distance of 8 m from the base of the wall. The child throws the stone from a height of 1 m with speed \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(20 ^ { \circ }\) above the horizontal.

1. Find the direction of motion of the stone when it hits the wall. The child now throws the stone with a speed of \(\mathrm { Vm } \mathrm { s } ^ { - 1 }\) from the same initial position and still at an angle of \(20 ^ { \circ }\) above the horizontal. This time the stone hits the target which is 2.5 m above the ground.
2. Find \(V\).

6 An athlete 'puts the shot' with an initial speed of \(19 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at an angle of \(11 ^ { \circ }\) above the horizontal. At the instant of release the shot is 1.53 m above the horizontal ground. By treating the shot as a particle and ignoring air resistance, find

1. the maximum height, above the ground, reached by the shot,
2. the horizontal distance the shot has travelled when it hits the ground.

3 A particle \(P\), of mass 2 kg , is initially at rest at a point \(O\) on a smooth horizontal surface. The particle moves along a straight line, \(O A\), under the action of a horizontal force. When the force has been acting for \(t\) seconds, it has magnitude \(( 4 t + 5 ) \mathrm { N }\).

1. Find the magnitude of the impulse exerted by the force on \(P\) between the times \(t = 0\) and \(t = 3\).
2. Find the speed of \(P\) when \(t = 3\).
3. The speed of \(P\) at \(A\) is \(37.5 \mathrm {~ms} ^ { - 1 }\). Find the time taken for the particle to reach \(A\).