Find power and resistance simultaneously

4 A lorry of mass 15000 kg moves on a straight horizontal road in the direction from \(A\) to \(B\). It passes \(A\) and \(B\) with speeds \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) respectively. The power of the lorry's engine is constant and there is a constant resistance to motion of magnitude 6000 N . The acceleration of the lorry at \(B\) is 0.5 times the acceleration of the lorry at \(A\).

1. Show that the power of the lorry's engine is 200 kW , and hence find the acceleration of the lorry when it is travelling at \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
   The lorry begins to ascend a straight hill inclined at \(1 ^ { \circ }\) to the horizontal. It is given that the power of the lorry's engine and the resistance force do not change.
2. Find the steady speed up the hill that the lorry could maintain.

2 A car of mass 1250 kg is travelling along a straight horizontal road with its engine working at a constant rate of \(P \mathrm {~W}\). The resistance to the car's motion is constant and equal to \(R \mathrm {~N}\). When the speed of the car is \(19 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) its acceleration is \(0.6 \mathrm {~m} \mathrm {~s} ^ { - 2 }\), and when the speed of the car is \(30 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) its acceleration is \(0.16 \mathrm {~m} \mathrm {~s} ^ { - 2 }\). Find the values of \(P\) and \(R\).

4 A car of mass 1230 kg increases its speed from \(4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(21 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in 24.5 s . The table below shows corresponding values of time \(t \mathrm {~s}\) and speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Using the values in the table, find the average acceleration of the car for \(0 < t < 0.5\) and for \(16.3 < t < 24.5\). While the car is increasing its speed the power output of its engine is constant and equal to \(P \mathrm {~W}\), and the resistance to the car's motion is constant and equal to \(R \mathrm {~N}\).
2. Assuming that the values obtained in part (i) are approximately equal to the accelerations at \(v = 5\) and at \(v = 20\), find approximations for \(P\) and \(R\).

2 A car of mass 600 kg travels along a horizontal straight road, with its engine working at a rate of 40 kW . The resistance to motion of the car is constant and equal to 800 N . The car passes through the point \(A\) on the road with speed \(25 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The car's acceleration at the point \(B\) on the road is half its acceleration at \(A\). Find the speed of the car at \(B\).

2. A vehicle of mass 450 kg is moving on a straight road that is inclined at angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 15 }\) At the instant when the vehicle is moving down the road at \(12 \mathrm {~ms} ^ { - 1 }\)

• the engine of the vehicle is working at a rate of \(P\) watts
• the acceleration of the vehicle is \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)
• the resistance to the motion of the vehicle is modelled as a constant force of magnitude \(R\) newtons

At the instant when the vehicle is moving up the road at \(12 \mathrm {~ms} ^ { - 1 }\)

• the engine of the vehicle is working at a rate of \(2 P\) watts
• the deceleration of the vehicle is \(0.5 \mathrm {~ms} ^ { - 2 }\)
• the resistance to the motion of the vehicle from non-gravitational forces is modelled as a constant force of magnitude \(R\) newtons

Find the value of \(P\).

3 A car of mass 1500 kg travels along a straight horizontal road with its engine working at a constant rate of \(P \mathrm {~W}\). There is a constant resistance to motion of \(R \mathrm {~N}\). Points \(A\) and \(B\) are on the road. At point \(A\) the car's speed is \(16 \mathrm {~ms} ^ { - 1 }\) and its acceleration is \(0.3875 \mathrm {~ms} ^ { - 2 }\). At point \(B\) the car's speed is \(25 \mathrm {~ms} ^ { - 1 }\) and its acceleration is \(0.2 \mathrm {~ms} ^ { - 2 }\). Find the values of \(P\) and \(R\).

6. A slope is inclined at an angle of \(5 ^ { \circ }\) to the horizontal. A car, of mass 1500 kg , has an engine that is working at a constant rate of \(P \mathrm {~W}\). The resistance to motion of the car is constant at 4500 N . When the car is moving up the slope, its acceleration is \(a \mathrm {~ms} ^ { - 2 }\) at the instant when its speed is \(10 \mathrm {~ms} ^ { - 1 }\). When the car is moving down the slope, its deceleration is \(a \mathrm {~ms} ^ { - 2 }\) at the instant when its speed is \(20 \mathrm {~ms} ^ { - 1 }\). Determine the value of \(P\) and the value of \(a\). \includegraphics[max width=\textwidth, alt={}, center]{ae23a093-1419-4be4-8285-951650dc5a35-14_87_1609_635_267}

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]