3.02d Constant acceleration: SUVAT formulae

In this question use \(g = 9.8\) m s\(^{-2}\) A ball is launched vertically upwards from the Earth's surface with velocity \(u\) m s\(^{-1}\) The ball reaches a maximum height of 15 metres. You may assume that air resistance can be ignored. Find the value of \(u\) [3 marks]

It is given that two points \(A\) and \(B\) have position vectors $$\overrightarrow{OA} = \begin{bmatrix} 5 \\ -1 \end{bmatrix} \text{ metres} \quad \text{and} \quad \overrightarrow{OB} = \begin{bmatrix} 13 \\ 5 \end{bmatrix} \text{ metres.}$$

1. Show that the distance from \(A\) to \(B\) is 10 metres. [3 marks]
2. A constant resultant force, of magnitude \(R\) newtons, acts on a particle so that it moves in a straight line passing through the same two points \(A\) and \(B\) At \(A\), the speed of the particle is 3 m s\(^{-1}\) in the direction from \(A\) to \(B\) The particle takes 2 seconds to travel from \(A\) to \(B\) The mass of the particle is 150 grams. Find the value of \(R\) [3 marks]

The graph shows how the speed of a cyclist varies during a timed section of length 120 metres along a straight track. \includegraphics{figure_15}

1. Find the acceleration of the cyclist during the first 10 seconds. [1 mark]
2. After the first 15 seconds, the cyclist travels at a constant speed of 5 m s⁻¹ for a further \(T\) seconds to complete the 120-metre section. Calculate the value of \(T\). [4 marks]

A garden snail moves in a straight line from rest to 1.28 cm s\(^{-1}\), with a constant acceleration in 1.8 seconds. Find the acceleration of the snail. Circle your answer. [1 mark] 2.30 m s\(^{-2}\) 0.71 m s\(^{-2}\) 0.0071 m s\(^{-2}\) 0.023 m s\(^{-2}\)

A driver is road-testing two minibuses, A and B, for a taxi company. The performance of each minibus along a straight track is compared. A flag is dropped to indicate the start of the test. Each minibus starts from rest. The acceleration in m s\(^{-2}\) of each minibus is modelled as a function of time, \(t\) seconds, after the flag is dropped: The acceleration of A = \(0.138 t^2\) The acceleration of B = \(0.024 t^3\)

1. Find the time taken for A to travel 100 metres. Give your answer to four significant figures. [4 marks]
2. The company decides to buy the minibus which travels 100 metres in the shortest time. Determine which minibus should be bought. [4 marks]
3. The models assume that both minibuses start moving immediately when \(t = 0\) In light of this, explain why the company may, in reality, make the wrong decision. [1 mark]

A buggy is pulling a roller-skater, in a straight line along a horizontal road, by means of a connecting rope as shown in the diagram. \includegraphics{figure_6} The combined mass of the buggy and driver is 410 kg A driving force of 300 N and a total resistance force of 140 N act on the buggy. The mass of the roller-skater is 72 kg A total resistance force of R newtons acts on the roller-skater. The buggy and the roller-skater have an acceleration of 0.2 m s\(^{-2}\)

1. 1. Find R. [3 marks]
   2. Find the tension in the rope. [3 marks]
2. State a necessary assumption that you have made. [1 mark]
3. The roller-skater releases the rope at a point A, when she reaches a speed of 6 m s\(^{-1}\) She continues to move forward, experiencing the same resistance force. The driver notices a change in motion of the buggy, and brings it to rest at a distance of 20 m from A.
   1. Determine whether the roller-skater will stop before reaching the stationary buggy. Fully justify your answer. [5 marks]
   2. Explain the change in motion that the driver noticed. [2 marks]

Two particles \(A\) and \(B\) are released from rest from different starting points above a horizontal surface. \(A\) is released from a height of \(h\) metres. \(B\) is released at a time \(t\) seconds after \(A\) from a height of \(kh\) metres, where \(0 < k < 1\) Both particles land on the surface \(5\) seconds after \(A\) was released. Assuming any resistance forces may be ignored, prove that $$t = 5(1 - \sqrt{k})$$ Fully justify your answer. [5 marks]

Block \(A\), of mass \(0.2\) kg, lies at rest on a rough plane. The plane is inclined at an angle \(\theta\) to the horizontal, such that \(\tan \theta = \frac{7}{24}\) A light inextensible string is attached to \(A\) and runs parallel to the line of greatest slope until it passes over a smooth fixed pulley at the top of the slope. The other end of this string is attached to particle \(B\), of mass \(2\) kg, which is held at rest so that the string is taut, as shown in the diagram below. \includegraphics{figure_18}

1. \(B\) is released from rest so that it begins to move vertically downwards with an acceleration of \(\frac{543}{625}\) g ms\(^{-2}\) Show that the coefficient of friction between \(A\) and the surface of the inclined plane is \(0.17\) [8 marks]
2. In this question use \(g = 9.81\text{ ms}^{-2}\) When \(A\) reaches a speed of \(0.5\text{ ms}^{-1}\) the string breaks.
   1. Find the distance travelled by \(A\) after the string breaks until first coming to rest. [4 marks]
   2. State an assumption that could affect the validity of your answer to part (b)(i). [1 mark]

In this question use \(g = 9.8\) m s\(^{-2}\) An apple tree stands on horizontal ground. An apple hangs, at rest, from a branch of the tree. A second apple also hangs, at rest, from a different branch of the tree. The vertical distance between the two apples is \(d\) centimetres. At the same instant both apples begin to fall freely under gravity. The first apple hits the ground after 0.5 seconds. The second apple hits the ground 0.1 seconds later. Show that \(d\) is approximately 54 [4 marks]

In this question use \(g = 9.8\) m s\(^{-2}\) A toy shoots balls upwards with an initial velocity of 7 m s\(^{-1}\) The advertisement for this toy claims the balls can reach a maximum height of 2.5 metres from the ground.

1. Suppose that the toy shoots the balls vertically upwards.
   1. Verify the claim in the advertisement. [2 marks]
   2. State two modelling assumptions you have made in verifying this claim. [2 marks]
2. In fact the toy shoots the balls anywhere between 0 and 11 degrees from the vertical. The range of maximum heights, \(h\) metres, above the ground which can be reached by the balls may be expressed as $$k \leq h \leq 2.5$$ Find the value of \(k\) [4 marks]

A particle moves on a straight line with a constant acceleration, \(a\) m s\(^{-2}\). The initial velocity of the particle is \(U\) m s\(^{-1}\). After \(T\) seconds the particle has velocity \(V\) m s\(^{-1}\). This information is shown on the velocity-time graph. \includegraphics{figure_12} The displacement, \(S\) metres, of the particle from its initial position at time \(T\) seconds is given by the formula $$S = \frac{1}{2}(U + V)T$$

1. By considering the gradient of the graph, or otherwise, write down a formula for \(a\) in terms of \(U\), \(V\) and \(T\). [1 mark]
2. Hence show that \(V^2 = U^2 + 2aS\) [3 marks]

In this question use \(g = 9.81\) m s\(^{-2}\). A ball is projected from the origin. After 2.5 seconds, the ball lands at the point with position vector \((40\mathbf{i} - 10\mathbf{j})\) metres. The unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal and vertical respectively. Assume that there are no resistance forces acting on the ball.

1. Find the speed of the ball when it is at a height of 3 metres above its initial position. [6 marks]
2. State the speed of the ball when it is at its maximum height. [1 mark]
3. Explain why the answer you found in part (b) may not be the actual speed of the ball when it is at its maximum height. [1 mark]

A car of mass 800kg travels up a line of greatest slope of a straight road inclined at \(5°\) to the horizontal. The power developed by the car is constant and equal to 25kW. The resistance to the motion of the car is constant and equal to 750N. The car passes through a point A on the road with speed \(7\)ms\(^{-1}\).

1. Find
   [5]

The car later passes through a point B on the road where AB = 131m. The time taken to travel from A to B is 10.4s.

1. Calculate the speed of the car at B. [6]

A particle P of mass 2 kg is projected vertically upwards from horizontal ground with an initial speed of \(14 \text{ m s}^{-1}\). At the same instant a particle Q of mass 8 kg is released from rest 5 m vertically above P. During the subsequent motion P and Q collide. The coefficient of restitution between P and Q is \(\frac{11}{14}\). Determine the time between this collision and P subsequently hitting the ground. [10]

A vehicle moves along a straight horizontal road. Points \(A\) and \(B\) lie on the road. As the vehicle passes point \(A\), it is moving with constant speed 15 ms\(^{-1}\). It travels with this constant speed for 2 minutes before a constant deceleration is applied for 12 seconds so that it comes to rest at point \(B\).

1. Find the distance \(AB\). [3]

The vehicle then reverses with a constant acceleration of 2 ms\(^{-2}\) for 8 seconds, followed by a constant deceleration of 1·6 ms\(^{-2}\), coming to rest at the point \(C\), which is between \(A\) and \(B\).

1. Calculate the time it takes for the vehicle to reverse from \(B\) to \(C\). [4]
2. Sketch a velocity-time graph for the motion of the vehicle. [3]
3. Determine the distance \(AC\). [2]

The diagram below shows an object \(A\), of mass \(2m\) kg, lying on a horizontal table. It is connected to another object \(B\), of mass \(m\) kg, by a light inextensible string, which passes over a smooth pulley \(P\), fixed at the edge of the table. Initially, object \(A\) is held at rest so that object \(B\) hangs freely with the string taut. \includegraphics{figure_9} Object \(A\) is then released.

1. When object \(B\) has moved downwards a vertical distance of 0·4 m, its speed is 1·2 ms\(^{-1}\). Use a formula for motion in a straight line with constant acceleration to show that the magnitude of the acceleration of \(B\) is 1·8 ms\(^{-2}\). [2]
2. During the motion, object \(A\) experiences a constant resistive force of 22 N. Find the value of \(m\) and hence determine the tension in the string. [6]
3. What assumption did the word 'inextensible' in the description of the string enable you to make in your solution? [1]

A car, starting from rest at a point \(A\), travels along a straight horizontal road towards a point \(B\). The distance between points \(A\) and \(B\) is 1·9 km. Initially, the car accelerates uniformly for 12 seconds until it reaches a speed of 26 ms\(^{-1}\). The car continues at 26 ms\(^{-1}\) for 1 minute, before decelerating at a constant rate of 0·75 ms\(^{-2}\) until it passes the point \(B\).

1. Show that the car travels 156 m while it is accelerating. [2]
2. 1. Work out the distance travelled by the car while travelling at a constant speed. [1]
   2. Hence calculate the length of time for which the car is decelerating until it passes the point \(B\). [5]
3. Sketch a displacement-time graph for the motion of the car between \(A\) and \(B\). [3]

A small object, of mass 0.02 kg, is dropped from rest from the top of a building which is160 m high.

1. Calculate the speed of the object as it hits the ground. [3]
2. Determine the time taken for the object to reach the ground. [3]
3. State one assumption you have made in your solution. [1]

A car of mass \(1250\) kg experiences a resistance to its motion of magnitude \(kv^2\) N, where \(k\) is a constant and \(v\) m s\(^{-1}\) is the car's speed. The car travels in a straight line along a horizontal road with its engine working at a constant rate of \(P\) W. At a point \(A\) on the road the car's speed is \(15\) m s\(^{-1}\) and it has an acceleration of magnitude \(0.54\) m s\(^{-2}\). At a point \(B\) on the road the car's speed is \(20\) m s\(^{-1}\) and it has an acceleration of magnitude \(0.3\) m s\(^{-2}\).

1. Find the values of \(k\) and \(P\). [7]

The power is increased to \(15\) kW.

1. Calculate the maximum steady speed of the car on a straight horizontal road. [3]

A car of mass \(600\)kg pulls a trailer of mass \(150\)kg along a straight horizontal road. The trailer is connected to the car by a light inextensible towbar, which is parallel to the direction of motion of the car. The resistance to the motion of the trailer is modelled as a constant force of magnitude \(200\)N. At the instant when the speed of the car is \(v\text{ms}^{-1}\), the resistance to the motion of the car is modelled as a force of magnitude \((200 + \lambda v)\)N, where \(\lambda\) is a constant. When the engine of the car is working at a constant rate of \(15\)kW, the car is moving at a constant speed of \(25\text{ms}^{-1}\).

1. Show that \(\lambda = 8\). [4]
2. Later on, the car is pulling the trailer up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin\theta = \frac{1}{15}\). The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude \(200\)N at all times. At the instant when the speed of the car is \(v\text{ms}^{-1}\), the resistance to the motion of the car from non-gravitational forces is modelled as a force of magnitude \((200 + 8v)\)N. The engine of the car is again working at a constant rate of \(15\)kW. When \(v = 10\), the towbar breaks. The trailer comes to instantaneous rest after moving a distance \(d\) metres up the road from the point where the towbar broke. Find the acceleration of the car immediately after the towbar breaks. [4]
3. Use the work-energy principle to find the value of \(d\). [4]

A car is initially travelling with a constant velocity of \(15 \text{ m s}^{-1}\) for \(T\) s. It then decelerates at a constant rate for \(\frac{T}{2}\) s, reaching a velocity of \(10 \text{ m s}^{-1}\). It then immediately accelerates at a constant rate for \(\frac{3T}{2}\) s reaching a velocity of \(20 \text{ m s}^{-1}\).

1. Sketch a velocity–time graph to illustrate the motion. [3]
2. Given that the car travels a total distance of 1312.5 m over the journey described, find the value of \(T\). [4]

A box \(A\) of mass 0.8 kg rests on a rough horizontal table and is attached to one end of a light inextensible string. The string passes over a smooth pulley fixed at the edge of the table. The other end of the string is attached to a sphere \(B\) of mass 1.2 kg, which hangs freely below the pulley. The magnitude of the frictional force between \(A\) and the table is \(F\) N. The system is released from rest when the string is taut. After release, \(B\) descends a distance of 0.9 m in 0.8 s. Modelling \(A\) and \(B\) as particles, calculate

1. the acceleration of \(B\), [2]
2. the tension in the string, [3]
3. the value of \(F\). [3]

A racing car starts from rest at the point \(A\) and moves with constant acceleration of \(11 \text{ m s}^{-2}\) for \(8 \text{ s}\). The velocity it has reached after \(8 \text{ s}\) is then maintained for \(7 \text{ s}\). The racing car then decelerates from this velocity to \(40 \text{ m s}^{-1}\) in a further \(2 \text{ s}\), reaching point \(B\).

1. Sketch a velocity-time graph to illustrate the motion of the racing car. Include the top speed of the racing car in your sketch. [5]
2. Given that the distance between \(A\) and \(B\) is \(1404 \text{ m}\), find the value of \(T\). [3]

A car of mass \(1200 \text{ kg}\) pulls a trailer of mass \(400 \text{ kg}\) along a straight horizontal road. The car and trailer are connected by a tow-rope modelled as a light inextensible rod. The engine of the car provides a constant driving force of \(3200 \text{ N}\). The horizontal resistances of the car and the trailer are proportional to their respective masses. Given that the acceleration of the car and the trailer is \(0.4 \text{ m s}^{-2}\),

1. find the resistance to motion on the trailer, [4]
2. find the tension in the tow-rope. [3]

When the car and trailer are travelling at \(25 \text{ m s}^{-1}\) the tow-rope breaks. Assuming that the resistances to motion remain unchanged,

1. find the distance the trailer travels before coming to a stop, [4]
2. state how you have used the modelling assumption that the tow-rope is inextensible. [1]

Answer all the questions. Two cyclists, \(A\) and \(B\), are cycling along the same straight horizontal track. The cyclists are modelled as particles and the motion of the cyclists is modelled as follows: • At time \(t = 0\), cyclist \(A\) passes through the point \(O\) with speed \(2\text{ms}^{-1}\) • Cyclist \(A\) is moving in a straight line with constant acceleration \(2\text{ms}^{-2}\) • At time \(t = 2\) seconds, cyclist \(B\) starts from rest at \(O\) • Cyclist \(B\) moves with constant acceleration \(6\text{ms}^{-2}\) along the same straight line and in the same direction as cyclist \(A\) • At time \(t = T\) seconds, \(B\) overtakes \(A\) at the point \(X\) Using the model,

1. sketch, on the same axes, for the interval from \(t = 0\) to \(t = T\) seconds, • a velocity-time graph for the motion of \(A\) • a velocity-time graph for the motion of \(B\) [2]
2. explain why the two graphs must cross before time \(t = T\) seconds, [1]
3. find the time when \(A\) and \(B\) are moving at the same speed, [2]
4. find the distance \(OX\) [5]