Engine power on road constant/variable speed

6. A car of mass 800 kg pulls a trailer of mass 300 kg up a straight road which is inclined to the horizontal at an angle \(\alpha\), where \(\sin \alpha = \frac { 1 } { 14 }\). The trailer is attached to the car by a light inextensible towbar which is parallel to the direction of motion of the car. The car's engine works at a constant rate of \(P \mathrm {~kW}\). The non-gravitational resistances to motion are constant and of magnitude 600 N on the car and 200 N on the trailer. At a given instant the car is moving at \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and is accelerating at \(0.5 \mathrm {~m} \mathrm {~s} ^ { - 2 }\)

1. Find the value of \(P\). When the car is moving up the road at \(12 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), 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. The non-gravitational resistance to the motion of the trailer remains constant and of magnitude 200 N .
2. Find, using the work-energy principle, the value of \(d\).
   

1. A cyclist and his bicycle have a total mass of 75 kg . The cyclist is moving down a straight road that is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 15 }\)

The cyclist is working at a constant rate of 56 W . The magnitude of the resistance to motion is modelled as a constant force of magnitude 40 N . At the instant when the speed of the cyclist is \(\mathrm { Vm } \mathrm { s } ^ { - 1 }\), his acceleration is \(\frac { 1 } { 3 } \mathrm {~ms} ^ { - 2 }\) Find the value of \(V\).
(5)

3. A car of mass 600 kg travels along a straight horizontal road with the engine of the car working at a constant rate of \(P\) watts. The resistance to the motion of the car is modelled as a constant force of magnitude \(R\) newtons. At the instant when the speed of the car is \(15 \mathrm {~ms} ^ { - 1 }\), the magnitude of the acceleration of the car is \(0.2 \mathrm {~ms} ^ { - 2 }\). Later the same car travels up a straight road inclined at angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 20 }\). The resistance to the motion of the car from non-gravitational forces is modelled as a constant force of magnitude \(R\) newtons. When the engine of the car is working at a constant rate of \(P\) watts, the car has a constant speed of \(10 \mathrm {~ms} ^ { - 1 }\). Find the value of \(P\).

2. A car of mass 600 kg tows a trailer of mass 200 kg up a hill along a straight road that is inclined at angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 20 }\). The trailer is attached to the car by a light inextensible towbar. The resistance to the motion of the car from non-gravitational forces is modelled as a constant force of magnitude 150 N . The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude 300 N . When the engine of the car is working at a constant rate of \(P \mathrm {~kW}\) the car and the trailer have a constant speed of \(15 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)

1. Find the value of \(P\). Later, at the instant when the car and the trailer are travelling up the hill with a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the towbar breaks. When the towbar breaks the trailer is at the point \(X\). The trailer continues to travel up the hill before coming to instantaneous rest at the point \(Y\). The resistance to the motion of the trailer from non-gravitational forces is again modelled as a constant force of magnitude 300 N .
2. Use the work-energy principle to find the distance \(X Y\).
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5. \begin{figure}[h] \end{figure} A van of mass 600 kg is moving up a straight road inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 14 }\). The van is towing a trailer of mass 200 kg . The trailer is attached to the van by a rigid towbar which is parallel to the direction of motion of the van and the trailer, as shown in Figure 4. The resistance to the motion of the van from non-gravitational forces is modelled as a constant force of magnitude 250 N . The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude 150 N . The towbar is modelled as a light rod.
At the instant when the speed of the van is \(16 \mathrm {~ms} ^ { - 1 }\), the engine of the van is working at a rate of 10 kW .

1. Find the deceleration of the van at this instant.
2. Find the tension in the towbar at this instant.

2. A car of mass 900 kg is moving down a straight road which is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 12 }\) The engine of the car is working at a constant rate of 15 kW .
The resistance to the motion of the car is modelled as a constant force of magnitude 400 N . Find the acceleration of the car at the instant when it is moving at \(16 \mathrm {~ms} ^ { - 1 }\) \section*{Qu}

2. A car of mass 800 kg is moving on a straight road which is inclined at an angle \(\theta\) to the horizontal, where \(\sin \theta = \frac { 1 } { 20 }\). The resistance to the motion of the car from non-gravitational forces is modelled as a constant force of magnitude \(R\) newtons. When the car is moving up the road at a constant speed of \(12.5 \mathrm {~ms} ^ { - 1 }\), the engine of the car is working at a constant rate of \(3 P\) watts. When the car is moving down the road at a constant speed of \(12.5 \mathrm {~ms} ^ { - 1 }\), the engine of the car is working at a constant rate of \(P\) watts.

1. Find
   1. the value of \(P\),
   2. the value of \(R\).
      (6) When the car is moving up the road at \(12.5 \mathrm {~ms} ^ { - 1 }\) the engine is switched off and the car comes to rest, without braking, in a distance \(d\) metres. The resistance to the motion of the car from non-gravitational forces is still modelled as a constant force of magnitude \(R\) newtons.
2. Use the work-energy principle to find the value of \(d\).

5 A car of mass 1500 kg travels up a line of greatest slope of a straight road inclined at \(5 ^ { \circ }\) to the horizontal. The power of the car's engine is constant and equal to 25 kW and the resistance to the motion of the car is constant and equal to 750 N . The car passes through point \(A\) with speed \(10 \mathrm {~ms} ^ { - 1 }\).

1. Find the acceleration of the car at \(A\). The car later passes through a point \(B\) with speed \(20 \mathrm {~ms} ^ { - 1 }\). The car takes 28s to travel from \(A\) to \(B\).
2. Find the distance \(A B\).

4 A particle \(B\) of mass 5 kg is at rest at the bottom of a slope which is angled at \(\sin ^ { - 1 } 0.2\) above the horizontal. A constant force \(D\) initially acts directly up the slope on \(B\). The total resistance to the motion of \(B\) is modelled as being a constant 12 N .
At the instant that \(D\) stops acting, the speed of \(B\) is \(18 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and \(B\) has moved 90 m up the slope.
Determine the average power of \(D\) over the time that \(D\) has been acting on \(B\).

6 A van, of mass 1400 kg , is accelerating at a constant rate of \(0.2 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) as it travels up a slope inclined at an angle \(\theta\) to the horizontal. The van experiences total resistance forces of 4000 N .
When the van is travelling at a speed of \(20 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), the power output of the van's engine is 91.1 kW . Find \(\theta\).
[0pt] [9 marks]

3 A rocket of mass 250 kg is moving in a straight line in space. There is no resistance to motion, and the mass of the rocket is assumed to be constant. With its motor working at a constant rate of 450 kW the rocket's speed increases from \(100 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(150 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) in a time \(t\) seconds.

1. Calculate the value of \(t\).
2. Calculate the acceleration of the rocket at the instant when its speed is \(120 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

5

1. A car of mass 800 kg is moving at a constant speed of \(20 \mathrm {~ms} ^ { - 1 }\) on a straight road down a hill inclined at an angle \(\alpha\) to the horizontal. The engine of the car works at a constant rate of 10 kW and there is a resistance to motion of 1300 N . Show that \(\sin \alpha = \frac { 5 } { 49 }\).
2. The car now travels up the same hill and its engine now works at a constant rate of 20 kW . The resistance to motion remains 1300 N . The car starts from rest and its speed is \(8 \mathrm {~ms} ^ { - 1 }\) after it has travelled a distance of 22.1 m . Calculate the time taken by the car to travel this distance.

1 A cyclist travels along a straight horizontal road. The total mass of the cyclist and her bicycle is 80 kg and the resistance to motion is a constant 60 N .

1. The cyclist travels at a constant speed working at a constant rate of 480 W . Find the speed at which she travels.
2. The cyclist now instantaneously increases her power to 600 W . After travelling at this power for 14.2 s her speed reaches \(9.4 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). Find the distance travelled at this power.

2. \begin{figure}[h] \end{figure} A van of mass 600 kg is moving up a straight road which is inclined at an angle \(\alpha\) to the horizontal, where \(\sin \alpha = \frac { 1 } { 15 }\). The van is towing a trailer of mass 150 kg . The van is attached to the trailer by a towbar which is parallel to the direction of motion of the van and the trailer, as shown in Figure 1. The resistance to the motion of the van from non-gravitational forces is modelled as a constant force of magnitude 200 N .
The resistance to the motion of the trailer from non-gravitational forces is modelled as a constant force of magnitude 100 N . The towbar is modelled as a light rod.
The engine of the van is working at a constant rate of 12 kW .
Find the tension in the towbar at the instant when the speed of the van is \(9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\)