3.02d Constant acceleration: SUVAT formulae

Small stones A and B are initially in the positions shown in Fig. 6 with B a height \(H\) m directly above A. \includegraphics{figure_5} At the instant when B is released from rest, A is projected vertically upwards with a speed of \(29.4\text{ms}^{-1}\). Air resistance may be neglected. The stones collide \(T\) seconds after they begin to move. At this instant they have the same speed, \(V\text{ms}^{-1}\), 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 \(H\). [7]

A particle \(A\) is released from rest from the top of a smooth plane, which makes an angle of 30° with the horizontal. The particle \(A\) collides 2 s later with a particle \(B\), which is moving up a line of greatest slope of the plane. The coefficient of restitution between the particles is 0.4 and the speed of \(B\) immediately before the collision is 2 ms\(^{-1}\). \(B\) has velocity 1 ms\(^{-1}\) down the plane immediately after the collision. Find

1. the speed of \(A\) immediately after the collision, [4]
2. the distance \(A\) moves up the plane after the collision. [2]

The masses of \(A\) and \(B\) are 0.5 kg and \(m\) kg, respectively.

1. Find the value of \(m\). [3]

A particle of mass 0.5 kg is held at rest at a point \(P\), which is at the bottom of an inclined plane. The particle is given an impulse of 1.8 N s directed up a line of greatest slope of the plane.

1. Find the speed at which the particle starts to move. [2]

The particle subsequently moves up the plane to a point \(Q\), which is 0.3 m above the level of \(P\).

1. Given that the plane is smooth, find the speed of the particle at \(Q\). [4]

It is given instead that the plane is rough. The particle is now projected up the plane from \(P\) with initial speed 3 ms\(^{-1}\), and comes to rest at a point \(R\) which is 0.2 m above the level of \(P\).

1. Given that the plane is inclined at 30° to the horizontal, find the magnitude of the frictional force on the particle. [4]

\includegraphics{figure_2} A small ball \(Q\) of mass \(2m\) is at rest at the point \(B\) on a smooth horizontal plane. A second small ball \(P\) of mass \(m\) is moving on the plane with speed \(\frac{13}{12}u\) and collides with \(Q\). Both the balls are smooth, uniform and of the same radius. The point \(C\) is on a smooth vertical wall \(W\) which is at a distance \(d_1\) from \(B\), and \(BC\) is perpendicular to \(W\). A second smooth vertical wall is perpendicular to \(W\) and at a distance \(d_2\) from \(B\). Immediately before the collision occurs, the direction of motion of \(P\) makes an angle \(\alpha\) with \(BC\), as shown in Fig. 2, where \(\tan \alpha = \frac{5}{12}\). The line of centres of \(P\) and \(Q\) is parallel to \(BC\). After the collision \(Q\) moves towards \(C\) with speed \(\frac{5}{4}u\).

1. Show that, after the collision, the velocity components of \(P\) parallel and perpendicular to \(CB\) are \(\frac{1}{4}u\) and \(\frac{5}{12}u\) respectively. [4]
2. Find the coefficient of restitution between \(P\) and \(Q\). [2]
3. Show that when \(Q\) reaches \(C\), \(P\) is at a distance \(\frac{4}{5}d_1\) from \(W\). [3]

For each collision between a ball and a wall the coefficient of restitution is \(\frac{1}{2}\). Given that the balls collide with each other again,

1. show that the time between the two collisions of the balls is \(\frac{15d_1}{u}\). [4]
2. find the ratio \(d_1 : d_2\). [5]

\includegraphics{figure_3} Figure 3 represents the scene of a road accident. A car of mass 600 kg collided at the point \(X\) with a stationary van of mass 800 kg. After the collision the van came to rest at the point \(A\) having travelled a horizontal distance of 45 m, and the car came to rest at the point \(B\) having travelled a horizontal distance of 21 m. The angle \(AXB\) is 90°. The accident investigators are trying to establish the speed of the car before the collision and they model both vehicles as small spheres.

1. Find the coefficient of restitution between the car and the van. [5]

The investigators assume that after the collision, and until the vehicles came to rest, the van was subject to a constant horizontal force of 500 N acting along \(AX\) and the car to a constant horizontal force of 300 N along \(BX\).

1. Find the speed of the car immediately before the collision. [9]

\includegraphics{figure_6} The diagram shows part of the curve \(y = \frac{2x - 1}{(2x + 3)(x + 1)^2}\). Find the exact area of the shaded region, giving your answer in the form \(p + q \ln r\), where \(p\) and \(q\) are positive integers and \(r\) is a positive rational number. [10]

A car is travelling on a straight horizontal road. The velocity of the car, \(v\) ms\(^{-1}\), at time \(t\) seconds as it travels past three points, \(P\), \(Q\) and \(R\), is modelled by the equation \(v = at^2 + bt + c\), where \(a\), \(b\) and \(c\) are constants. The car passes \(P\) at time \(t = 0\) with velocity \(8\) ms\(^{-1}\).

1. State the value of \(c\). [1] The car passes \(Q\) at time \(t = 5\) and at that instant its deceleration is \(0.12\) ms\(^{-2}\). The car passes \(R\) at time \(t = 18\) with velocity \(2.96\) ms\(^{-1}\).
2. Determine the values of \(a\) and \(b\). [4]
3. Find, to the nearest metre, the distance between points \(P\) and \(R\). [2]

There are three checkpoints, \(A\), \(B\) and \(C\), in that order, on a straight horizontal road. A car travels along the road, in the direction from \(A\) to \(C\), with constant acceleration. The car takes 20 s to travel from \(B\) to \(C\). The speed of the car at \(B\) is 14 m s\(^{-1}\) and the speed of the car at \(C\) is 18 m s\(^{-1}\).

1. Find the acceleration of the car. [1]

It is given that the distance between \(A\) and \(B\) is 330 m.

1. Determine the speed of the car at \(A\). [2]

\includegraphics{figure_9} The diagram shows a velocity-time graph representing the motion of two cars \(A\) and \(B\) which are both travelling along a horizontal straight road. At time \(t = 0\), car \(B\), which is travelling with constant speed \(12 \mathrm{m s}^{-1}\), is overtaken by car \(A\) which has initial speed \(20 \mathrm{m s}^{-1}\). From \(t = 0\) car \(A\) travels with constant deceleration for 30 seconds. When \(t = 30\) the speed of car \(A\) is \(8 \mathrm{m s}^{-1}\) and the car maintains this speed in subsequent motion.

1. Calculate the deceleration of car \(A\). [2]
2. Determine the value of \(t\) when \(B\) overtakes \(A\). [4]

In this question you should take the acceleration due to gravity to be \(10 \text{ms^{-2}\).} \includegraphics{figure_12} A small ball \(P\) is projected from a point \(A\) with speed \(39 \text{ms}^{-1}\) at an angle of elevation \(\theta\), where \(\sin \theta = \frac{5}{13}\) and \(\cos \theta = \frac{12}{13}\). Point \(A\) is \(20 \text{m}\) vertically above a point \(B\) on horizontal ground. The ball first lands at a point \(C\) on the horizontal ground (see diagram). The ball \(P\) is modelled as a particle moving freely under gravity.

1. Find the maximum height of \(P\) above the ground during its motion. [3]

The time taken for \(P\) to travel from \(A\) to \(C\) is \(7\) seconds.

1. Determine the value of \(T\). [3]
2. State one limitation of the model, other than air resistance or the wind, that could affect the answer to part (b). [1]

At the instant that \(P\) is projected, a second small ball \(Q\) is released from rest at \(B\) and moves towards \(C\) along the horizontal ground. At time \(t\) seconds, where \(t \geq 0\), the velocity \(v \text{ms}^{-1}\) of \(Q\) is given by $$v = kt^3 + 6t^2 + \frac{3}{2}t,$$ where \(k\) is a positive constant.

1. Given that \(P\) and \(Q\) collide at \(C\), determine the acceleration of \(Q\) immediately before this collision. [6]

\includegraphics{figure_13} The diagram shows a small block \(B\), of mass \(2 \text{kg}\), and a particle \(P\), of mass \(4 \text{kg}\), which are attached to the ends of a light inextensible string. The string is taut and passes over a small smooth pulley fixed at the intersection of a horizontal surface and an inclined plane. The particle can move on the inclined plane, which is rough, and which makes an angle of \(60°\) with the horizontal. The block can move on the horizontal surface, which is also rough. The system is released from rest, and in the subsequent motion \(P\) moves down the plane and \(B\) does not reach the pulley. It is given that the coefficient of friction between \(P\) and the inclined plane is twice the coefficient of friction between \(B\) and the horizontal surface.

1. Determine, in terms of \(g\), the tension in the string. [7]

When \(P\) is moving at \(2 \text{ms}^{-1}\) the string breaks. In the \(0.5\) seconds after the string breaks \(P\) moves \(1.9 \text{m}\) down the plane.

1. Determine the deceleration of \(B\) after the string breaks. Give your answer correct to 3 significant figures. [5]

In this question use \(g = 9.8\,\mathrm{m}\,\mathrm{s}^{-2}\) A ball, initially at rest, is dropped from a height of \(40\,\mathrm{m}\) above the ground. Calculate the speed of the ball when it reaches the ground. Circle your answer. [1 mark] \(-28\,\mathrm{m}\,\mathrm{s}^{-1}\) \quad \(28\,\mathrm{m}\,\mathrm{s}^{-1}\) \quad \(-780\,\mathrm{m}\,\mathrm{s}^{-1}\) \quad \(780\,\mathrm{m}\,\mathrm{s}^{-1}\)

A vehicle, which begins at rest at point \(P\), is travelling in a straight line. For the first \(4\) seconds the vehicle moves with a constant acceleration of \(0.75\,\mathrm{m}\,\mathrm{s}^{-2}\) For the next \(5\) seconds the vehicle moves with a constant acceleration of \(-1.2\,\mathrm{m}\,\mathrm{s}^{-2}\) The vehicle then immediately stops accelerating, and travels a further \(33\,\mathrm{m}\) at constant speed.

1. Draw a velocity-time graph for this journey on the grid below. [3 marks] \includegraphics{figure_13}
2. Find the distance of the car from \(P\) after \(20\) seconds. [3 marks]

A remote-controlled toy car is moving over a horizontal surface. It moves in a straight line through a point \(A\). The toy is initially at the point with displacement \(3\) metres from \(A\). Its velocity, \(v\,\mathrm{m}\,\mathrm{s}^{-1}\), at time \(t\) seconds is defined by $$v = 0.06(2 + t - t^2)$$

1. Find an expression for the displacement, \(r\) metres, of the toy from \(A\) at time \(t\) seconds. [4 marks]
2. In this question use \(g = 9.8\,\mathrm{m}\,\mathrm{s}^{-2}\) At time \(t = 2\) seconds, the toy launches a ball which travels directly upwards with initial speed \(3.43\,\mathrm{m}\,\mathrm{s}^{-1}\) Find the time taken for the ball to reach its highest point. [3 marks]

A ball moves in a straight line and passes through two fixed points, \(A\) and \(B\), which are \(0.5 \text{m}\) apart. The ball is moving with a constant acceleration of \(0.39 \text{m s}^{-2}\) in the direction \(AB\). The speed of the ball at \(A\) is \(1.9 \text{m s}^{-1}\) Find the speed of the ball at \(B\). Circle your answer. [1 mark] \(2 \text{m s}^{-1}\) \(3.2 \text{m s}^{-1}\) \(3.8 \text{m s}^{-1}\) \(4 \text{m s}^{-1}\)

Two particles, \(A\) and \(B\), lie at rest on a smooth horizontal plane. \(A\) has position vector \(\mathbf{r}_A = (13\mathbf{i} - 22\mathbf{j})\) metres \(B\) has position vector \(\mathbf{r}_B = (3\mathbf{i} + 2\mathbf{j})\) metres

1. Calculate the distance between \(A\) and \(B\). [2 marks]
2. Three forces, \(\mathbf{F}_1\), \(\mathbf{F}_2\) and \(\mathbf{F}_3\) are applied to particle \(A\), where \(\mathbf{F}_1 = (-2\mathbf{i} + 4\mathbf{j})\) newtons \(\mathbf{F}_2 = (6\mathbf{i} - 10\mathbf{j})\) newtons Given that \(A\) remains at rest, explain why \(\mathbf{F}_3 = (-4\mathbf{i} + 6\mathbf{j})\) newtons [1 mark]
3. A force of \((5\mathbf{i} - 12\mathbf{j})\) newtons, is applied to \(B\), so that \(B\) moves, from rest, in a straight line towards \(A\). \(B\) has a mass of \(0.8 \text{kg}\)
   1. Show that the acceleration of \(B\) towards \(A\) is \(16.25 \text{m s}^{-2}\) [2 marks]
   2. Hence, find the time taken for \(B\) to reach \(A\). Give your answer to two significant figures. [2 marks]

A tractor and its driver have a combined mass of \(m\) kilograms. The tractor is towing a trailer of mass \(4m\) kilograms in a straight line along a horizontal road. The tractor and trailer are connected by a horizontal tow bar, modelled as a light rigid rod. A driving force of \(11080 \text{N}\) and a total resistance force of \(160 \text{N}\) act on the tractor. A total resistance force of \(600 \text{N}\) acts on the trailer. The tractor and the trailer have an acceleration of \(0.8 \text{m s}^{-2}\)

1. Find \(m\). [3 marks]
2. Find the tension in the tow bar. [2 marks]
3. At the instant the speed of the tractor reaches \(18 \text{km h}^{-1}\) the tow bar breaks. The total resistance force acting on the trailer remains constant. Starting from the instant the tow bar breaks, calculate the time taken until the speed of the trailer reduces to \(9 \text{km h}^{-1}\) [4 marks]

An object is moving in a straight line, with constant acceleration \(a\text{ m s}^{-2}\), over a time period of \(t\) seconds. It has an initial velocity \(u\) and final velocity \(v\) as shown in the graph below. \includegraphics{figure_13} Use the graph to show that $$v = u + at$$ [3 marks]

A particle of mass 0.1 kg is initially stationary. A single force \(\mathbf{F}\) acts on this particle in a direction parallel to the vector \(7\mathbf{i} + 24\mathbf{j}\) As a result, the particle accelerates in a straight line, reaching a speed of \(4\text{ m s}^{-1}\) after travelling a distance of 3.2 m Find \(\mathbf{F}\). [5 marks]

A simple lifting mechanism comprises a light inextensible wire which is passed over a smooth fixed pulley. One end of the wire is attached to a rigid triangular container of mass 2 kg, which rests on horizontal ground. A load of \(m\) kg is placed in the container. The other end of the wire is attached to a particle of mass 5 kg, which hangs vertically downwards. The mechanism is initially held at rest as shown in the diagram below. \includegraphics{figure_16} The mechanism is released from rest, and the container begins to move upwards with acceleration \(a\text{ m s}^{-2}\) The wire remains taut throughout the motion.

1. Show that $$a = \left(\frac{3 - m}{m + 7}\right)g$$ [4 marks]
2. State the range of possible values of \(m\). [1 mark]
3. In this question use \(g = 9.8\text{ m s}^{-2}\) The load reaches a height of 2 metres above the ground 1 second after it is released. Find the mass of the load. [4 marks]
4. Ignoring air resistance, describe one assumption you have made in your model. [1 mark]

A particle P lies at rest on a smooth horizontal table. A constant resultant force, F newtons, is then applied to P. As a result P moves in a straight line with constant acceleration \(\begin{bmatrix}8\\6\end{bmatrix}\) m s⁻²

1. Show that the magnitude of the acceleration of P is 10 m s⁻² [1 mark]
2. Find the speed of P after 3 seconds. [1 mark]
3. Given that \(\mathbf{F} = \begin{bmatrix}2\\1.5\end{bmatrix}\) N, find the mass of P. [2 marks]

In this question, use \(g = 10\) m s⁻² A box, B, of mass 4 kg lies at rest on a fixed rough horizontal shelf. One end of a light string is connected to B. The string passes over a smooth peg, attached to the end of the shelf. The other end of the string is connected to particle, P, of mass 1 kg, which hangs freely below the shelf as shown in the diagram below. \includegraphics{figure_15} B is initially held at rest with the string taut. B is then released. B and P both move with constant acceleration \(a\) m s⁻² As B moves across the shelf it experiences a total resistance force of 5 N

1. State one type of force that would be included in the total resistance force. [1 mark]
2. Show that \(a = 1\) [4 marks]
3. When B has moved forward exactly 20 cm the string breaks. Find how much further B travels before coming to rest. [4 marks]
4. State one assumption you have made when finding your solutions in parts (b) or (c). [1 mark]

Jermaine and his friend Meena are walking in the same direction along a straight path. Meena is walking at a constant speed of \(u\) m s\(^{-1}\) Jermaine is walking 0.2 m s\(^{-1}\) more slowly than Meena. When Jermaine is \(d\) metres behind Meena he starts to run with a constant acceleration of 2 m s\(^{-2}\), for a time of \(t\) seconds, until he reaches her.

1. Show that $$d = t^2 - 0.2t$$ [4 marks]
2. When Jermaine's speed is 7.8 m s\(^{-1}\), he reaches Meena. Given that \(u = 1.4\) find the value of \(d\). [2 marks]

A particle, initially at rest, starts to move forward in a straight line with constant acceleration, \(a \text{ m s}^{-2}\) After 6 seconds the particle has a velocity of \(3 \text{ m s}^{-1}\) Find the value of \(a\) Circle your answer. [1 mark] \(-2\) \quad \(-0.5\) \quad \(0.5\) \quad \(2\)

A particle is moving in a straight line with constant acceleration \(a\) m s\(^{-2}\) The particle's velocity, \(v\) m s\(^{-1}\), varies with time, \(t\) seconds, so that $$v = 3 - 4t$$ Deduce the value of \(a\) Circle your answer. [1 mark] \(-4\) \qquad \(-1\) \qquad \(3\) \qquad \(4\)