6.02l Power and velocity: P = Fv

Relative to a fixed origin \(O\), the position vector \(\mathbf{r}\) m at time \(t\) s of a particle \(P\), of mass 0.4 kg, is given by $$\mathbf{r} = e^{2t}\mathbf{i} + \sin(2t)\mathbf{j} + \cos(2t)\mathbf{k}.$$

1. Show that the velocity vector \(\mathbf{v}\) and the position vector \(\mathbf{r}\) are never perpendicular to each other. [6]
2. Given that the speed of \(P\) at time \(t\) is \(v\) ms\(^{-1}\), show that $$v^2 = 4e^{4t} + 4.$$ [2]
3. Find the kinetic energy of \(P\) at time \(t\). [1]
4. Calculate the work done by the force acting on \(P\) in the interval \(0 < t < 1\). [2]
5. Determine an expression for the rate at which the force acting on \(P\) is working at time \(t\). [2]

A vehicle of mass 6000 kg is moving up a slope inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac{6}{49}\). The vehicle's engine exerts a constant power of \(P\) W. The constant resistance to motion of the vehicle is \(R\) N. At the instant the vehicle is moving with velocity \(\frac{16}{5}\) ms\(^{-1}\), its acceleration is 2 ms\(^{-2}\). The maximum velocity of the vehicle is \(\frac{16}{3}\) ms\(^{-1}\). Determine the value of \(P\) and the value of \(R\). [9]

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 of mass 800 kg is driven with its engine generating a power of 15 kW.

1. The car is first driven along a straight horizontal road and accelerates from rest. Assuming that there is no resistance to motion, find the speed of the car after 6 seconds. [2]
2. The car is next driven at constant speed up a straight road inclined at an angle \(\theta\) to the horizontal. The resistance to motion is now modelled as being constant with magnitude of 150 N. Given that \(\sin \theta = \frac{1}{20}\), find the speed of the car. [3]
3. The car is now driven at a constant speed of 30 ms\(^{-1}\) along the horizontal road pulling a trailer of mass 150 kg which is attached by means of a light rigid horizontal towbar. Assuming the resistance to motion of the car is three times the resistance to motion of the trailer. Find:
   1. the resistance to motion of the car,
   2. the magnitude of the tension in the towbar
   [4]

A van of mass 600 kg is moving up a straight road inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac{1}{16}\). The resistance to motion of the van from non-gravitational forces has constant magnitude \(R\) newtons. When the van is moving at a constant speed of 20 m s\(^{-1}\), the van's engine is working at a constant rate of 25 kW.

1. Find the value of \(R\). [4]

The power developed by the van's engine is now increased to 30 kW. The resistance to motion from non-gravitational forces is unchanged. At the instant when the van is moving up the road at 20 m s\(^{-1}\), the acceleration of the van is \(a\) m s\(^{-2}\).

1. Find the value of \(a\). [4]

A car of mass 1200 kg is driven on a long straight horizontal road. There is a constant force of 250 N resisting the motion of the car. The engine of the car is working at a constant power of 10 kW.

1. The car can travel at constant speed \(v \text{ ms}^{-1}\) along the road. Find \(v\). [2]
2. Find the acceleration of the car at an instant when its speed is \(30 \text{ ms}^{-1}\). [3]

A car of mass 850 kg is being driven uphill along a straight road inclined at \(7°\) to the horizontal. The resistance to motion is modelled as a constant force of magnitude 140 N. At a certain instant the car's speed is \(12 \text{ms}^{-1}\) and its acceleration is \(0.4 \text{ms}^{-2}\).

1. Calculate the power of the car's engine at this instant. [3]
2. Find the constant speed at which the car could travel up the hill with the engine generating this power. [2]

An engine is travelling along a straight horizontal track against negligible resistances. In travelling a distance of 750 m its speed increases from 5 m s\(^{-1}\) to 15 m s\(^{-1}\). Find the time taken if the engine was

1. exerting a constant tractive force, [2]
2. working at constant power. [9]