Energy method with work done

5 A cyclist and his bicycle have a total mass of 90 kg . The cyclist starts to move with speed \(3 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) from the top of a straight hill, of length 500 m , which is inclined at an angle of \(\sin ^ { - 1 } 0.05\) to the horizontal. The cyclist moves with constant acceleration until he reaches the bottom of the hill with speed \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The cyclist generates 420 W of power while moving down the hill. The resistance to the motion of the cyclist and his bicycle, \(R \mathrm {~N}\), and the cyclist's speed, \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\), both vary.

1. Show that \(R = \frac { 420 } { v } + 43.56\).
2. Find the cyclist's speed at the mid-point of the hill. Hence find the decrease in the value of \(R\) when the cyclist moves from the top of the hill to the mid-point of the hill, and when the cyclist moves from the mid-point of the hill to the bottom of the hill.

2. A bullet of mass 6 grams passes horizontally through a fixed, vertical board. After the bullet has travelled 2 cm through the board its speed is reduced from \(400 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(250 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The board exerts a constant resistive force on the bullet. Find, to 3 significant figures, the magnitude of this resistive force.
(5)

1 A bullet of mass 0.2 kg is fired into a fixed vertical barrier. It enters the barrier horizontally with speed \(250 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) and emerges horizontally after a time \(T\) seconds with speed \(40 \mathrm {~m} \mathrm {~s} ^ { - 1 }\). There is a constant horizontal resisting force of magnitude 1200 N . Find \(T\).

2 A bullet of mass 9 grams passes horizontally through a fixed vertical board of thickness 3 cm . The speed of the bullet is reduced from \(250 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(150 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) as it passes through the board. The board exerts a constant resistive force on the bullet. Calculate the magnitude of this resistive force.

2. A car of mass 1000 kg , moving along a straight horizontal road, is driven by an engine which produces a constant power of 12 kW . The only resistance to the motion of the car is air resistance of magnitude \(10 v ^ { 2 } \mathrm {~N}\) where \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) is the speed of the car. Find the distance travelled by the car as its speed increases from \(5 \mathrm {~m} \mathrm {~s} ^ { - 1 }\) to \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).
(8 marks)

1 A sports car of mass 1.2 tonnes is being tested on a horizontal, straight section of road. After \(t \mathrm {~s}\), the car has travelled \(x \mathrm {~m}\) from the starting line and its velocity is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\). The engine produces a driving force of 4000 N and the total resistance to the motion of the car is given by \(\frac { 40 } { 49 } v ^ { 2 } \mathrm {~N}\). The car crosses the starting line with speed \(10 \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. Write down an equation of motion for the car and solve it to show that \(v ^ { 2 } = 4900 - 4800 \mathrm { e } ^ { - \frac { 1 } { 735 } x }\).
2. Hence find the work done against the resistance to motion over the first 100 m beyond the starting line.

1 A car of mass \(m\) moves horizontally in a straight line. At time \(t\) the car is a distance \(x\) from a point O and is moving away from O with speed \(v\). There is a force of magnitude \(k v ^ { 2 }\), where \(k\) is a constant, resisting the motion of the car. The car's engine has a constant power \(P\). The terminal speed of the car is \(U\).

1. Show that $$m v ^ { 2 } \frac { \mathrm {~d} v } { \mathrm {~d} x } = P \left( 1 - \frac { v ^ { 3 } } { U ^ { 3 } } \right)$$
2. Show that the distance moved while the car accelerates from a speed of \(\frac { 1 } { 4 } U\) to a speed of \(\frac { 1 } { 2 } U\) is $$\frac { m U ^ { 3 } } { 3 P } \ln A$$ stating the exact value of the constant \(A\). Once the car attains a speed of \(\frac { 1 } { 2 } U\), no further power is supplied by the car's engine.
3. Find, in terms of \(m , P\) and \(U\), the time taken for the speed of the car to reduce from \(\frac { 1 } { 2 } U\) to \(\frac { 1 } { 4 } U\).