3.02d Constant acceleration: SUVAT formulae

\includegraphics{figure_3} Figure 3 shows a boat \(B\) of mass \(400\) kg held at rest on a slipway by a rope. The boat is modelled as a particle and the slipway as a rough plane inclined at \(15°\) to the horizontal. The coefficient of friction between \(B\) and the slipway is \(0.2\). The rope is modelled as a light, inextensible string, parallel to a line of greatest slope of the plane. The boat is in equilibrium and on the point of sliding down the slipway.

1. Calculate the tension in the rope. [6]

The boat is \(50\) m from the bottom of the slipway. The rope is detached from the boat and the boat slides down the slipway.

1. Calculate the time taken for the boat to slide to the bottom of the slipway. [6]

A small boat \(S\), drifting in the sea, is modelled as a particle moving in a straight line at constant speed. When first sighted at 0900, \(S\) is at a point with position vector \((4\mathbf{i} - 6\mathbf{j})\) km relative to a fixed origin \(O\), where \(\mathbf{i}\) and \(\mathbf{j}\) are unit vectors due east and due north respectively. At 0945, \(S\) is at the point with position vector \((7\mathbf{i} - 7.5\mathbf{j})\) km. At time \(t\) hours after 0900, \(S\) is at the point with position vector \(\mathbf{s}\) km.

1. Calculate the bearing on which \(S\) is drifting. [4]
2. Find an expression for \(\mathbf{s}\) in terms of \(t\). [3]

At 1000 a motor boat \(M\) leaves \(O\) and travels with constant velocity \((p\mathbf{i} + q\mathbf{j})\) km h\(^{-1}\). Given that \(M\) intercepts \(S\) at 1015,

1. calculate the value of \(p\) and the value of \(q\). [6]

\includegraphics{figure_4} Two particles \(P\) and \(Q\), of mass \(4\) kg and \(6\) kg respectively, are joined by a light inextensible string. Initially the particles are at rest on a rough horizontal plane with the string taut. The coefficient of friction between each particle and the plane is \(\frac{2}{5}\). A constant force of magnitude \(40\) N is then applied to \(Q\) in the direction \(PQ\), as shown in Fig. 4.

1. Show that the acceleration of \(Q\) is \(1.2\) m s\(^{-2}\). [4]
2. Calculate the tension in the string when the system is moving. [3]
3. State how you have used the information that the string is inextensible. [1]

After the particles have been moving for \(7\) s, the string breaks. The particle \(Q\) remains under the action of the force of magnitude \(40\) N.

1. Show that \(P\) continues to move for a further \(3\) seconds. [5]
2. Calculate the speed of \(Q\) at the instant when \(P\) comes to rest. [4]

In taking off, an aircraft moves on a straight runway \(AB\) of length 1.2 km. The aircraft moves from \(A\) with initial speed \(2 \text{ m s}^{-1}\). It moves with constant acceleration and 20 s later it leaves the runway at \(C\) with speed \(74 \text{ m s}^{-1}\). Find

1. the acceleration of the aircraft, [2]
2. the distance \(BC\). [4]

A train is travelling at \(10 \text{ m s}^{-1}\) on a straight horizontal track. The driver sees a red signal 135 m ahead and immediately applies the brakes. The train immediately decelerates with constant deceleration for 12 s, reducing its speed to \(3 \text{ m s}^{-1}\). The driver then releases the brakes and allows the train to travel at a constant speed of \(3 \text{ m s}^{-1}\) for a further 15 s. He then applies the brakes again and the train slows down with constant deceleration, coming to rest as it reaches the signal.

1. Sketch a speed-time graph to show the motion of the train, [3]
2. Find the distance travelled by the train from the moment when the brakes are first applied to the moment when its speed first reaches \(3 \text{ m s}^{-1}\). [2]
3. Find the total time from the moment when the brakes are first applied to the moment when the train comes to rest. [5]

\includegraphics{figure_4} Figure 4 shows a lorry of mass 1600 kg towing a car of mass 900 kg along a straight horizontal road. The two vehicles are joined by a light towbar which is at an angle of \(15°\) to the road. The lorry and the car experience constant resistances to motion of magnitude 600 N and 300 N respectively. The lorry's engine produces a constant horizontal force on the lorry of magnitude 1500 N. Find

1. the acceleration of the lorry and the car, [3]
2. the tension in the towbar. [4]

When the speed of the vehicles is \(6 \text{ m s}^{-1}\), the towbar breaks. Assuming that the resistance to the motion of the car remains of constant magnitude 300 N,

1. find the distance moved by the car from the moment the towbar breaks to the moment when the car comes to rest. [4]
2. State whether, when the towbar breaks, the normal reaction of the road on the car is increased, decreased or remains constant. Give a reason for your answer. [2]

[In this question, the unit vectors \(\mathbf{i}\) and \(\mathbf{j}\) are horizontal vectors due east and north respectively.] At time \(t = 0\), a football player kicks a ball from the point \(A\) with position vector \((2\mathbf{i} + \mathbf{j})\) m on a horizontal football field. The motion of the ball is modelled as that of a particle moving horizontally with constant velocity \((5\mathbf{i} + 8\mathbf{j}) \text{ m s}^{-1}\). Find

1. the speed of the ball, [2]
2. the position vector of the ball after \(t\) seconds. [2]

The point \(B\) on the field has position vector \((10\mathbf{i} + 7\mathbf{j})\) m.

1. Find the time when the ball is due north of \(B\). [2]

At time \(t = 0\), another player starts running due north from \(B\) and moves with constant speed \(v \text{ m s}^{-1}\). Given that he intercepts the ball,

1. find the value of \(v\). [6]
2. State one physical factor, other than air resistance, which would be needed in a refinement of the model of the ball's motion to make the model more realistic. [1]

Three posts \(P\), \(Q\) and \(R\) are fixed in that order at the side of a straight horizontal road. The distance from \(P\) to \(Q\) is 45 m and the distance from \(Q\) to \(R\) is 120 m. A car is moving along the road with constant acceleration \(a\) m s\(^{-2}\). The speed of the car, as it passes \(P\), is \(u\) m s\(^{-1}\). The car passes \(Q\) two seconds after passing \(P\), and the car passes \(R\) four seconds after passing \(Q\). Find

1. the value of \(u\),
2. the value of \(a\).

[7]

Two cars \(P\) and \(Q\) are moving in the same direction along the same straight horizontal road. Car \(P\) is moving with constant speed 25 m s\(^{-1}\). At time \(t = 0\), \(P\) overtakes \(Q\) which is moving with constant speed 20 m s\(^{-1}\). From \(t = T\) seconds, \(P\) decelerates uniformly, coming to rest at a point \(X\) which is 800 m from the point where \(P\) overtook \(Q\). From \(t = 25\) s, \(Q\) decelerates uniformly, coming to rest at the same point \(X\) at the same instant as \(P\).

1. Sketch, on the same axes, the speed-time graphs of the two cars for the period from \(t = 0\) to the time when they both come to rest at the point \(X\). [4]
2. Find the value of \(T\). [8]

At time \(t = 0\) a ball is projected vertically upwards from a point \(O\) and rises to a maximum height of 40 m above \(O\). The ball is modelled as a particle moving freely under gravity.

1. Show that the speed of projection is 28 m s\(^{-1}\). [3]
2. Find the times, in seconds, when the ball is 33.6 m above \(O\). [5]

A girl runs a 400 m race in a time of 84 s. In a model of this race, it is assumed that, starting from rest, she moves with constant acceleration for 4 s, reaching a speed of 5 m s\(^{-1}\). She maintains this speed for 60 s and then moves with constant deceleration for 20 s, crossing the finishing line with a speed of \(V\) m s\(^{-1}\).

1. Sketch, in the space below, a speed-time graph for the motion of the girl during the whole race. [2]
2. Find the distance run by the girl in the first 64 s of the race. [3]
3. Find the value of \(V\). [5]
4. Find the deceleration of the girl in the final 20 s of her race. [2]

\includegraphics{figure_2} Two particles \(P\) and \(Q\) have masses 0.3 kg and \(m\) kg respectively. The particles are attached to the ends of a light inextensible string. The string passes over a small smooth pulley which is fixed at the top of a fixed rough plane. The plane is inclined to the horizontal at an angle \(\alpha\), where \(\tan \alpha = \frac{3}{4}\). The coefficient of friction between \(P\) and the plane is \(\frac{1}{2}\). The string lies in a vertical plane through a line of greatest slope of the inclined plane. The particle \(P\) is held at rest on the inclined plane and the particle \(Q\) hangs freely below the pulley with the string taut, as shown in Figure 2. The system is released from rest and \(Q\) accelerates vertically downwards at 1.4 m s\(^{-2}\). Find

1. the magnitude of the normal reaction of the inclined plane on \(P\), [2]
2. the value of \(m\). [8]

When the particles have been moving for 0.5 s, the string breaks. Assuming that \(P\) does not reach the pulley,

1. find the further time that elapses until \(P\) comes to instantaneous rest. [6]

At time \(t = 0\), two balls \(A\) and \(B\) are projected vertically upwards. The ball \(A\) is projected vertically upwards with speed 2 m s\(^{-1}\) from a point 50 m above the horizontal ground. The ball \(B\) is projected vertically upwards from the ground with speed 20 m s\(^{-1}\). At time \(t = T\) seconds, the two balls are at the same vertical height, \(h\) metres, above the ground. The balls are modelled as particles moving freely under gravity. Find

1. the value of \(T\), [5]
2. the value of \(h\). [2]

\includegraphics{figure_3} A particle \(P\) of mass 0.6 kg slides with constant acceleration down a line of greatest slope of a rough plane, which is inclined at 25° to the horizontal. The particle passes through two points \(A\) and \(B\), where \(AB = 10\) m, as shown in Figure 3. The speed of \(P\) at \(A\) is 2 m s\(^{-1}\). The particle \(P\) takes 3.5 s to move from \(A\) to \(B\). Find

1. the speed of \(P\) at \(B\), [3]
2. the acceleration of \(P\), [2]
3. the coefficient of friction between \(P\) and the plane. [5]

\includegraphics{figure_4} A truck of mass 1750 kg is towing a car of mass 750 kg along a straight horizontal road. The two vehicles are joined by a light towbar which is inclined at an angle \(\theta\) to the road, as shown in Figure 4. The vehicles are travelling at 20 m s\(^{-1}\) as they enter a zone where the speed limit is 14 m s\(^{-1}\). The truck's brakes are applied to give a constant braking force on the truck. The distance travelled between the instant when the brakes are applied and the instant when the speed of each vehicle is 14 m s\(^{-1}\) is 100 m.

1. Find the deceleration of the truck and the car. [3]

The constant braking force on the truck has magnitude \(R\) newtons. The truck and the car also experience constant resistances to motion of 500 N and 300 N respectively. Given that cos \(\theta = 0.9\), find

1. the force in the towbar, [4]
2. the value of \(R\). [4]

A lorry is moving along a straight horizontal road with constant acceleration. The lorry passes a point \(A\) with speed \(u \text{ m s}^{-1}\), \((u < 34)\), and 10 seconds later passes a point \(B\) with speed \(34 \text{ m s}^{-1}\). Given that \(AB = 240\) m, find

1. the value of \(u\), [3]
2. the time taken for the lorry to move from \(A\) to the mid-point of \(AB\). [6]

A car is travelling along a straight horizontal road. The car takes 120 s to travel between two sets of traffic lights which are 2145 m apart. The car starts from rest at the first set of traffic lights and moves with constant acceleration for 30 s until its speed is \(22 \text{ m s}^{-1}\). The car maintains this speed for \(T\) seconds. The car then moves with constant deceleration, coming to rest at the second set of traffic lights.

1. Sketch, in the space below, a speed-time graph for the motion of the car between the two sets of traffic lights. [2]
2. Find the value of \(T\). [3]

A motorcycle leaves the first set of traffic lights 10 s after the car has left the first set of traffic lights. The motorcycle moves from rest with constant acceleration, \(a \text{ m s}^{-2}\), and passes the car at the point \(A\) which is 990 m from the first set of traffic lights. When the motorcycle passes the car, the car is moving with speed \(22 \text{ m s}^{-1}\).

1. Find the time it takes for the motorcycle to move from the first set of traffic lights to the point \(A\). [4]
2. Find the value of \(a\). [2]

An aircraft moves along a straight horizontal runway with constant acceleration. It passes a point \(A\) on the runway with speed \(16\) m s\(^{-1}\). It then passes the point \(B\) on the runway with speed \(34\) m s\(^{-1}\). The distance from \(A\) to \(B\) is \(150\) m.

1. Find the acceleration of the aircraft. [3]
2. Find the time taken by the aircraft in moving from \(A\) to \(B\). [2]
3. Find, to 3 significant figures, the speed of the aircraft when it passes the point mid-way between \(A\) and \(B\). [2]

A racing car is travelling on a straight horizontal road. Its initial speed is \(25\) m s\(^{-1}\) and it accelerates for \(4\) s to reach a speed of \(V\) m s\(^{-1}\). It then travels at a constant speed of \(V\) m s\(^{-1}\) for a further \(8\) s. The total distance travelled by the car during this \(12\) s period is \(600\) m.

1. Sketch a speed-time graph to illustrate the motion of the car during this \(12\) s period. [2]
2. Find the value of \(V\). [4]
3. Find the acceleration of the car during the initial \(4\) s period. [2]

A small ball is projected vertically upwards from a point A. The greatest height reached by the ball is 40 m above A. Calculate

1. the speed of projection. [3]
2. the time between the instant that the ball is projected and the instant it returns to A. [3]

A car starts from rest at a point \(S\) on a straight racetrack. The car moves with constant acceleration for 20 s, reaching a speed of 25 m s\(^{-1}\). The car then travels at a constant speed of 25 m s\(^{-1}\) for 120 s. Finally it moves with constant deceleration, coming to rest at a point \(F\).

1. In the space below, sketch a speed-time graph to illustrate the motion of the car. [2]

The distance between \(S\) and \(F\) is 4 km.

1. Calculate the total time the car takes to travel from \(S\) to \(F\). [3]

A motorcycle starts at \(S\), 10 s after the car has left \(S\). The motorcycle moves with constant acceleration from rest and passes the car at a point \(P\) which is 1.5 km from \(S\). When the motorcycle passes the car, the motorcycle is still accelerating and the car is moving at a constant speed. Calculate

1. the time the motorcycle takes to travel from \(S\) to \(P\), [5]
2. the speed of the motorcycle at \(P\). [2]

\includegraphics{figure_3} Figure 3 shows two particles \(A\) and \(B\), of mass \(m\) kg and 0.4 kg respectively, connected by a light inextensible string. Initially \(A\) is held at rest on a fixed smooth plane inclined at 30° to the horizontal. The string passes over a small light smooth pulley \(P\) fixed at the top of the plane. The section of the string from \(A\) to \(P\) is parallel to a line of greatest slope of the plane. The particle \(B\) hangs freely below \(P\). The system is released from rest with the string taut and \(B\) descends with acceleration \(\frac{1}{8}g\).

1. Write down an equation of motion for \(B\). [2]
2. Find the tension in the string. [2]
3. Prove that \(m = \frac{16}{35}\). [4]
4. State where in the calculations you have used the information that \(P\) is a light smooth pulley. [1]

On release, \(B\) is at a height of one metre above the ground and \(AP = 1.4\) m. The particle \(B\) strikes the ground and does not rebound.

1. Calculate the speed of \(B\) as it reaches the ground. [2]
2. Show that \(A\) comes to rest as it reaches \(P\). [5]

END

A man is driving a car on a straight horizontal road. He sees a junction \(S\) ahead, at which he must stop. When the car is at the point \(P\), 300 m from \(S\), its speed is \(30 \text{ m s}^{-1}\). The car continues at this constant speed for 2 s after passing \(P\). The man then applies the brakes so that the car has constant deceleration and comes to rest at \(S\).

1. Sketch, in the space below, a speed-time graph to illustrate the motion of the car in moving from \(P\) to \(S\). [2]
2. Find the time taken by the car to travel from \(P\) to \(S\). [3]

Two cars \(A\) and \(B\) are moving in the same direction along a straight horizontal road. At time \(t = 0\), they are side by side, passing a point \(O\) on the road. Car \(A\) travels at a constant speed of \(30 \text{ m s}^{-1}\). Car \(B\) passes \(O\) with a speed of \(20 \text{ m s}^{-1}\), and has constant acceleration of \(4 \text{ m s}^{-2}\). Find

1. the speed of \(B\) when it has travelled 78 m from \(O\), [2]
2. the distance from \(O\) of \(A\) when \(B\) is 78 m from \(O\), [4]
3. the time when \(B\) overtakes \(A\). [5]

A car starts from rest at a point \(O\) and moves in a straight line. The car moves with constant acceleration \(4 \text{ m s}^{-2}\) until it passes the point \(A\) when it is moving with speed \(10 \text{ m s}^{-1}\). It then moves with constant acceleration \(3 \text{ m s}^{-2}\) for 6 s until it reaches the point \(B\). Find

1. the speed of the car at \(B\), [2]
2. the distance \(OB\). [5]