3.02d Constant acceleration: SUVAT formulae

2 A particle travels with constant acceleration along a straight line. A and B are points on this line 8 m apart. The motion of the particle is as follows.

• Initially it is at A.
• After 32 s it is at B .
• When it is at B its speed is \(2.25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and it is moving away from A .

In either order, calculate the acceleration and the initial velocity of the particle, making the directions clear.

5 A small object is projected over horizontal ground from a point O at ground level and makes a loud noise on landing. It has an initial speed of \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) at \(35 ^ { \circ }\) to the horizontal. Assuming that air resistance on the object may be neglected and that the speed of sound in air is \(343 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), calculate how long after projection the noise is heard at O .

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 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 .

6 A football is kicked with speed \(31 \mathrm {~ms} ^ { - 1 }\) 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).

2 In this question, air resistance should be neglected.
Fig. 2 illustrates the flight of a golf ball. The golf ball is initially on the ground, which is horizontal. \begin{figure}[h] \end{figure} It is hit and given an initial velocity with components of \(15 \mathrm {~ms} ^ { - 1 }\) in the horizontal direction and \(20 \mathrm {~ms} ^ { - 1 }\) in the vertical direction.

1. Find its initial speed.
2. Find the ball's flight time and range, \(R \mathrm {~m}\).
3. (A) Show that the range is the same if the components of the initial velocity of the ball are \(20 \mathrm {~ms} ^ { - 1 }\) in the horizontal direction and \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in the vertical direction.
   (B) State, justifying your answer, whether the range is the same whenever the ball is hit with the same initial speed.

1 A particle \(P\) is projected vertically downwards with initial speed \(3.5 \mathrm {~ms} ^ { - 1 }\) from a point \(A\) which is 5 m above horizontal ground.

1. Find the speed of \(P\) immediately before it strikes the ground. After striking the ground, \(P\) rebounds and moves vertically upwards and 0.87 s after leaving the ground \(P\) passes through \(A\).
2. Calculate the speed of \(P\) immediately after it leaves the ground. It is given that the mass of \(P\) is 0.2 kg .
3. Calculate the change in the momentum of \(P\) as a result of its collision with the ground.

3 A car travels along a straight road with constant acceleration \(a \mathrm {~ms} ^ { - 2 }\), where \(a > 0\). The car passes through points \(A , B\) and \(C\) in that order. The speed of the car at \(A\) is \(u \mathrm {~ms} ^ { - 1 }\) in the direction \(A B\). The distance \(B C\) is twice the distance \(A B\). The car takes 8 seconds to travel from \(A\) to \(B\) and 10 seconds to travel from \(B\) to \(C\).

1. Find \(u\) in terms of \(a\).
2. Find the speed of the car at \(C\) in terms of \(a\).

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.
   (3 marks)
2. Find her speed when the parachute opens.
   (2 marks)
   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.
   (5 marks)
   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 particle of mass \(m \mathrm {~kg}\) is projected directly up a rough plane with a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The plane makes an angle of \(30 ^ { \circ }\) with the horizontal and the coefficient of friction is 0.2 . Calculate the distance the particle travels up the plane before coming instantaneously to rest.

6 A car of mass 700 kg is travelling up a hill which is inclined at a constant angle of \(5 ^ { \circ }\) to the horizontal. At a certain point \(P\) on the hill the car's speed is \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The point \(Q\) is 400 m further up the hill from \(P\), and at \(Q\) the car's speed is \(15 \mathrm {~ms} ^ { - 1 }\).

1. Calculate the work done by the car's engine as the car moves from \(P\) to \(Q\), assuming that any resistances to the car's motion may be neglected. Assume instead that the resistance to the car's motion between \(P\) and \(Q\) is a constant force of magnitude 200 N.
2. Given that the acceleration of the car at \(Q\) is zero, show that the power of the engine as the car passes through \(Q\) is 12.0 kW , correct to 3 significant figures.
3. Given that the power of the car's engine at \(P\) is the same as at \(Q\), calculate the car's retardation at \(P\).

7 A small ball is projected at an angle of \(50 ^ { \circ }\) above the horizontal, from a point \(A\), which is 2 m above ground level. The highest point of the path of the ball is 15 m above the ground, which is horizontal. Air resistance may be ignored.

1. Find the speed with which the ball is projected from \(A\). The ball hits a net at a point \(B\) when it has travelled a horizontal distance of 45 m .
2. Find the height of \(B\) above the ground.
3. Find the speed of the ball immediately before it hits the net.

11 The velocity \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) of a car at time \(t \mathrm {~s}\), during the first 20 s of its journey, is given by \(v = k t + 0.03 t ^ { 2 }\), where \(k\) is a constant. When \(t = 20\) the acceleration of the car is \(1.3 \mathrm {~ms} ^ { - 2 }\). For \(t > 20\) the car continues its journey with constant acceleration \(1.3 \mathrm {~ms} ^ { - 2 }\) until its speed reaches \(25 \mathrm {~ms} ^ { - 1 }\).

1. Find the value of \(k\).
2. Find the total distance the car has travelled when its speed reaches \(25 \mathrm {~ms} ^ { - 1 }\).

1. At time \(t = 0\), a parachutist falls vertically from rest from a helicopter which is hovering at a height of 550 m above horizontal ground.

The parachutist, who is modelled as a particle, falls for 3 seconds before her parachute opens.
While she is falling, and before her parachute opens, she is modelled as falling freely under gravity. The acceleration due to gravity is modelled as being \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\).

1. Using this model, find the speed of the parachutist at the instant her parachute opens. When her parachute is open, the parachutist continues to fall vertically.
   Immediately after her parachute opens, she decelerates at \(12 \mathrm {~ms} ^ { - 2 }\) for 2 seconds before reaching a constant speed and she reaches the ground with this speed. The total time taken by the parachutist to fall the 550 m from the helicopter to the ground is \(T\) seconds.
2. Sketch a speed-time graph for the motion of the parachutist for \(0 \leqslant t \leqslant T\).
3. Find, to the nearest whole number, the value of \(T\). In a refinement of the model of the motion of the parachutist, the effect of air resistance is included before her parachute opens and this refined model is now used to find a new value of \(T\).
4. How would this new value of \(T\) compare with the value found, using the initial model, in part (c)?
5. Suggest one further refinement to the model, apart from air resistance, to make the model more realistic.

2. \begin{figure}[h] \end{figure} A small ball, \(P\), of mass 0.8 kg , is held at rest on a smooth horizontal table and is attached to one end of a thin rope. The rope passes over a pulley that is fixed at the edge of the table.
The other end of the rope is attached to another small ball, \(Q\), of mass 0.6 kg , that hangs freely below the pulley. Ball \(P\) is released from rest, with the rope taut, with \(P\) at a distance of 1.5 m from the pulley and with \(Q\) at a height of 0.4 m above the horizontal floor, as shown in Figure 1. Ball \(Q\) descends, hits the floor and does not rebound.
The balls are modelled as particles, the rope as a light and inextensible string and the pulley as small and smooth. Using this model,

1. show that the acceleration of \(Q\), as it falls, is \(4.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
2. find the time taken by \(P\) to hit the pulley from the instant when \(P\) is released.
3. State one limitation of the model that will affect the accuracy of your answer to part (a).

1. At time \(t = 0\), a small ball is projected vertically upwards with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from a point \(A\) that is 16.8 m above horizontal ground.

The speed of the ball at the instant immediately before it hits the ground for the first time is \(19 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The ball hits the ground for the first time at time \(t = T\) seconds.
The motion of the ball, from the instant it is projected until the instant just before it hits the ground for the first time, is modelled as that of a particle moving freely under gravity. The acceleration due to gravity is modelled as having magnitude \(10 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) Using the model,

1. show that \(U = 5\)
2. find the value of \(T\),
3. find the time from the instant the ball is projected until the instant when the ball is 1.2 m below \(A\).
4. Sketch a velocity-time graph for the motion of the ball for \(0 \leqslant t \leqslant T\), stating the coordinates of the start point and the end point of your graph. In a refinement of the model of the motion of the ball, the effect of air resistance on the ball is included and this refined model is now used to find the value of \(U\).
5. State, with a reason, how this new value of \(U\) would compare with the value found in part (a), using the initial unrefined model.
6. Suggest one further refinement that could be made to the model, apart from including air resistance, that would make the model more realistic.

1. The point \(A\) is 1.8 m vertically above horizontal ground.

At time \(t = 0\), a small stone is projected vertically upwards with speed \(U \mathrm {~ms} ^ { - 1 }\) from the point \(A\). At time \(t = T\) seconds, the stone hits the ground.
The speed of the stone as it hits the ground is \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) In an initial model of the motion of the stone as it moves from \(A\) to where it hits the ground

• the stone is modelled as a particle moving freely under gravity
• the acceleration due to gravity is modelled as having magnitude \(\mathbf { 1 0 } \mathbf { m ~ s } ^ { \mathbf { - 2 } }\)

Using the model,

1. find the value of \(U\),
2. find the value of \(T\).
3. Suggest one refinement, apart from including air resistance, that would make the model more realistic. In reality the stone will not move freely under gravity and will be subject to air resistance.
4. Explain how this would affect your answer to part (a).

1. A train travels along a straight horizontal track from station \(P\) to station \(Q\).

In a model of the motion of the train, at time \(t = 0\) the train starts from rest at \(P\), and moves with constant acceleration until it reaches its maximum speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The train then travels at this constant speed of \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) before finally moving with constant deceleration until it comes to rest at \(Q\). The time spent decelerating is four times the time spent accelerating.
The journey from \(P\) to \(Q\) takes 700 s .
Using the model,

1. sketch a speed-time graph for the motion of the train between the two stations \(P\) and \(Q\). The distance between the two stations is 15 km .
   Using the model,
2. show that the time spent accelerating by the train is 40 s ,
3. find the acceleration, in \(\mathrm { m } \mathrm { s } ^ { - 2 }\), of the train,
4. find the speed of the train 572s after leaving \(P\).
5. State one limitation of the model which could affect your answers to parts (b) and (c).

1. \begin{figure}[h] \end{figure} Two children, Pat \(( P )\) and Sam \(( S )\), run a race along a straight horizontal track.
Both children start from rest at the same time and cross the finish line at the same time.
In a model of the motion:
Pat accelerates at a constant rate from rest for 5 s until reaching a speed of \(4 \mathrm {~ms} ^ { - 1 }\) and then maintains a constant speed of \(4 \mathrm {~ms} ^ { - 1 }\) until crossing the finish line. Sam accelerates at a constant rate of \(1 \mathrm {~ms} ^ { - 2 }\) from rest until reaching a speed of \(X \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and then maintains a constant speed of \(X \mathrm {~m} \mathrm {~s} ^ { - 1 }\) until crossing the finish line. Both children take 27.5 s to complete the race.
The velocity-time graphs shown in Figure 1 describe the model of the motion of each child from the instant they start to the instant they cross the finish line together. Using the model,

1. explain why the areas under the two graphs are equal,
2. find the acceleration of Pat during the first 5 seconds,
3. find, in metres, the length of the race,
4. find the value of \(X\), giving your answer to 3 significant figures.

1. A small stone is projected vertically upwards with speed \(39.2 \mathrm {~ms} ^ { - 1 }\) from a point \(O\).

The stone is modelled as a particle moving freely under gravity from when it is projected until it hits the ground 10s later. Using the model, find

1. the height of \(O\) above the ground,
2. the total length of time for which the speed of the stone is less than or equal to \(24.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)
3. State one refinement that could be made to the model that would make your answer to part (a) more accurate.

1. \begin{figure}[h] \end{figure} Figure 1 shows the speed-time graph for the journey of a car moving in a long queue of traffic on a straight horizontal road. At time \(\mathrm { t } = 0\), the car is at rest at the point A .
The car then accelerates uniformly for 5 seconds until it reaches a speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) For the next 15 seconds the car travels at a constant speed of \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) The car then decelerates uniformly until it comes to rest at the point B.
The total journey time is 30 seconds.

1. Find the distance AB .
2. Sketch a distance-time graph for the journey of the car from A to B .

4. \begin{figure}[h] \end{figure} Figure 2 shows a car towing a trailer along a straight horizontal road.
The mass of the car is 800 kg and the mass of the trailer is 600 kg .
The trailer is attached to the car by a towbar which is parallel to the road and parallel to the direction of motion of the car and the trailer. The towbar is modelled as a light rod.
The resistance to the motion of the car is modelled as a constant force of magnitude 400 N .
The resistance to the motion of the trailer is modelled as a constant force of magnitude R newtons. The engine of the car is producing a constant driving force that is horizontal and of magnitude 1740 N. The acceleration of the car is \(0.6 \mathrm {~ms} ^ { - 2 }\) and the tension in the towbar is T newtons.
Using the model,

1. show that \(\mathrm { R } = 500\)
2. find the value of T . At the instant when the speed of the car and the trailer is \(12.5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the towbar breaks.
   The trailer moves a further distance d metres before coming to rest.
   The resistance to the motion of the trailer is modelled as a constant force of magnitude 500 N. Using the model,
3. show that, after the towbar breaks, the deceleration of the trailer is \(\frac { 5 } { 6 } \mathrm {~ms} ^ { - 2 }\)
4. find the value of d. In reality, the distance d metres is likely to be different from the answer found in part (d).
5. Give two different reasons why this is the case.

1. At time \(t = 0\), a small stone is thrown vertically upwards with speed \(14.7 \mathrm {~ms} ^ { - 1 }\) from a point \(A\).

At time \(t = T\) seconds, the stone passes through \(A\), moving downwards.
The stone is modelled as a particle moving freely under gravity throughout its motion.
Using the model,

1. find the value of \(T\),
2. find the total distance travelled by the stone in the first 4 seconds of its motion.
3. State one refinement that could be made to the model, apart from air resistance, that would make the model more realistic.

1. A car is initially at rest on a straight horizontal road.

The car then accelerates along the road with a constant acceleration of \(3.2 \mathrm {~ms} ^ { - 2 }\) Find

1. the speed of the car after 5 s ,
2. the distance travelled by the car in the first 5 s .

2. \begin{figure}[h] \end{figure} Figure 2 shows a speed-time graph for a model of the motion of an athlete running a \(\mathbf { 2 0 0 m }\) race in 24 s . The athlete

• starts from rest at time \(t = 0\) and accelerates at a constant rate, reaching a speed of \(10 \mathrm {~ms} ^ { - 1 }\) at \(t = 4\)
• then moves at a constant speed of \(10 \mathrm {~ms} ^ { - 1 }\) from \(t = 4\) to \(t = 18\)
• then decelerates at a constant rate from \(t = 18\) to \(t = 24\), crossing the finishing line with speed \(U \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

Using the model,

1. find the acceleration of the athlete during the first 4 s of the race, stating the units of your answer,
2. find the distance covered by the athlete during the first 18s of the race,
3. find the value of \(U\).

10 Particles \(P\) and \(Q\), of masses 3 kg and 5 kg respectively, are attached to the ends of a light inextensible string. The string passes over a smooth fixed pulley. The system is held at rest with the string taut. The hanging parts of the string are vertical and \(P\) and \(Q\) are above a horizontal plane (see diagram). \includegraphics[max width=\textwidth, alt={}, center]{efde7b10-b4f3-469f-ba91-b765a16ea835-6_428_208_932_932}

1. Find the tension in the string immediately after the particles are released. After descending \(2.5 \mathrm {~m} , Q\) strikes the plane and is immediately brought to rest. It is given that \(P\) does not reach the pulley in the subsequent motion.
2. Find the distance travelled by \(P\) between the instant when \(Q\) strikes the plane and the instant when the string becomes taut again.