Bungee jumping problems

4. A man of mass 75 kg is attached to one end of a light elastic rope of natural length 12 m . The other end of the rope is attached to a point on the edge of a horizontal ledge 19 m above the ground. The man steps off the ledge and falls vertically under gravity. The man is modelled as a particle falling from rest. He is brought to instantaneous rest by the rope when he is 1 m above the ground.
Find

1. the modulus of elasticity of the rope,
   (5)
2. the speed of the man when he is 2 m above the ground, giving your answer in \(\mathrm { m } \mathrm { s } ^ { - 1 }\) to 3 significant figures.
   (5)

2 A bungee jumper of mass 80 kg steps off a high bridge with an elastic rope attached to her ankles. She is assumed to fall vertically from rest and the air resistance she experiences is modelled as a constant force of 32 N . The rope has natural length 4 m and modulus of elasticity 470 N . By considering energy, determine the total distance she falls before first coming to instantaneous rest.

8 Carol, a bungee jumper of mass 70 kg , is attached to one end of a light elastic cord of natural length 26 metres and modulus of elasticity 1456 N . The other end of the cord is attached to a fixed horizontal platform which is at a height of 69 metres above the ground. Carol steps off the platform at the point where the cord is attached and falls vertically. Hooke's law can be assumed to apply whilst the cord is taut. Model Carol as a particle and assume air resistance to be negligible.
When Carol has fallen \(x \mathrm {~m}\), her speed is \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\).

1. By considering energy, show that $$5 v ^ { 2 } = 306 x - 4 x ^ { 2 } - 2704 \text { for } x \geqslant 26$$
2. Why is the expression found in part (a) not true when \(x\) takes values less than 26?
3. Find the maximum value of \(x\).
4. 1. Find the distance fallen by Carol when her speed is a maximum.
   2. Hence find Carol's maximum speed.

6 A bungee jumper of mass 70 kg is joined to a fixed point \(O\) by a light elastic rope of natural length 30 m and modulus of elasticity 1470 N . The jumper starts from rest at \(O\) and falls vertically. The jumper is modelled as a particle and air resistance is ignored.

1. Find the distance fallen by the jumper when maximum speed is reached.
2. Show that this maximum speed is \(26.9 \mathrm {~m} \mathrm {~s} ^ { - 1 }\), correct to 3 significant figures.
3. Find the extension of the rope when the jumper is at the lowest position. \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{09d3e8ca-0062-4f62-8453-7acaff591db5-4_543_616_310_301} \captionsetup{labelformat=empty} \caption{Fig. 1}
   \end{figure} \begin{figure}[h]
   \includegraphics[alt={},max width=\textwidth]{09d3e8ca-0062-4f62-8453-7acaff591db5-4_668_709_267_1135} \captionsetup{labelformat=empty} \caption{Fig. 2}
   \end{figure} A smooth horizontal cylinder of radius 0.6 m is fixed with its axis horizontal and passing through a fixed point \(O\). A light inextensible string of length \(0.6 \pi \mathrm {~m}\) has particles \(P\) and \(Q\), of masses 0.3 kg and 0.4 kg respectively, attached at its ends. The string passes over the cylinder and is held at rest with \(P , O\) and \(Q\) in a straight horizontal line (see Fig. 1). The string is released and \(Q\) begins to descend. When the line \(O P\) makes an angle \(\theta\) radians, \(0 \leqslant \theta \leqslant \frac { 1 } { 2 } \pi\), with the horizontal, the particles have speed \(v \mathrm {~m} \mathrm {~s} ^ { - 1 }\) (see Fig. 2).
4. By considering the total energy of the system, or otherwise, show that $$v ^ { 2 } = 6.72 \theta - 5.04 \sin \theta .$$
5. Show that the magnitude of the contact force between \(P\) and the cylinder is $$( 5.46 \sin \theta - 3.36 \theta ) \text { newtons. }$$ Hence find the value of \(\theta\) for which the magnitude of the contact force is greatest.
6. Find the transverse component of the acceleration of \(P\) in terms of \(\theta\).

4 For a bungee jump, a girl is joined to a fixed point \(O\) of a bridge by an elastic rope of natural length 25 m and modulus of elasticity 1320 N . The girl starts from rest at \(O\) and falls vertically. The lowest point reached by the girl is 60 m vertically below \(O\). The girl is modelled as a particle, the rope is assumed to be light, and air resistance is neglected.

1. Find the greatest tension in the rope during the girl's jump.
2. Use energy considerations to find
   1. the mass of the girl,
   2. the speed of the girl when she has fallen half way to the lowest point.

3 Ben has mass 60 kg and he is considering doing a bungee jump using an elastic rope with natural length 32 m . One end of the rope is attached to a fixed point O , and the other end is attached to Ben. When Ben is supported in equilibrium by the rope, the length of the rope is 32.8 m . To predict what will happen, Ben is modelled as a particle B, the rope is assumed to be light, and air resistance is neglected. B is released from rest at O and falls vertically. When the rope becomes stretched, \(x \mathrm {~m}\) denotes the extension of the rope.

1. Find the stiffness of the rope.
2. Use an energy argument to show that, when B comes to rest instantaneously with the rope stretched, $$x ^ { 2 } - 1.6 x - 51.2 = 0$$ Hence find the length of the rope when B is at its lowest point.
3. Show that, while the rope is stretched, $$\frac { \mathrm { d } ^ { 2 } x } { \mathrm {~d} t ^ { 2 } } + 12.25 x = 9.8$$ where \(t\) is the time measured in seconds.
4. Find the time taken for B to travel between the equilibrium position \(( x = 0.8 )\) and the lowest point.
5. Find the acceleration of \(\mathbf { B }\) when it is at the lowest point, and comment on the implications for Ben.

3 A young woman wishes to make a bungee jump. One end of an elastic rope is attached to her safety harness. The other end is attached to the bridge from which she will jump. She calculates that the stretched length of the rope at the bottom of her motion should be 20 m , she knows that her weight is 576 N and the stiffness of the elastic rope is \(90 \mathrm { Nm } ^ { - 1 }\). She has to calculate the unstretched length of rope required to perform the jump safely. She models the situation by assuming the following.

• The rope is of negligible mass.
• Air resistance may be neglected.
• She is a particle.
• She moves vertically downwards from rest.
• Her starting point is level with the fixed end of the rope.
• The length she calculates for the rope does not include any extra for attaching the ends.
  1. (A) Show that the greatest extension of the rope, \(X\), satisfies the equation \(X ^ { 2 } = 256\).
     (B) Hence determine the natural length of rope she needs.
  2. To remain safe she wishes to be sure that, if air resistance is taken into account, the stretched length of the rope of natural length determined in part (i) will not be more than 20 m . Advise her on this point.

9 A bungee jumper, of mass 80 kg , is attached to one end of a light elastic cord, of natural length 16 metres and modulus of elasticity 784 N . The other end of the cord is attached to a horizontal platform, which is at a height of 65 metres above the ground. The bungee jumper steps off the platform at the point where the cord is attached and falls vertically. The bungee jumper can be modelled as a particle. Hooke's law can be assumed to apply throughout the motion and air resistance can be assumed to be negligible.

1. Find the length of the cord when the acceleration of the bungee jumper is zero.
2. The cord extends by \(x\) metres beyond its natural length before the bungee jumper first comes to rest.
   1. Show that \(x ^ { 2 } - 32 x - 512 = 0\).
   2. Find the distance above the ground at which the bungee jumper first comes to rest.

8 A bungee jumper, of mass 49 kg , is attached to one end of a light elastic cord of natural length 22 metres and modulus of elasticity 1078 newtons. The other end of the cord is attached to a horizontal platform, which is at a height of 60 metres above the ground. The bungee jumper steps off the platform at the point where the cord is attached, and falls vertically. The bungee jumper can be modelled as a particle. Assume that Hooke's Law applies whilst the cord is taut and that air resistance is negligible throughout the motion. When the bungee jumper has fallen \(x\) metres, his speed is \(v \mathrm {~ms} ^ { - 1 }\).

1. By considering energy, show that, when \(x\) is greater than 22, $$5 v ^ { 2 } = 318 x - 5 x ^ { 2 } - 2420$$
2. Explain why \(x\) must be greater than 22 for the equation in part (a) to be valid. ( 1 mark)
3. Find the maximum value of \(x\).
4. 1. Show that the speed of the bungee jumper is a maximum when \(x = 31.8\).
   2. Hence find the maximum speed of the bungee jumper.

8 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) Omar, a bungee jumper of mass 70 kg , has his ankles attached to one end of an elastic cord. The other end of the cord is attached to a bridge which is 80 metres above the surface of a river. Omar steps off the bridge at the point where the cord is attached and falls vertically downwards. The cord can be modelled as a light elastic string of natural length \(L\) metres and modulus of elasticity 2800 N Model Omar as a particle. 8

1. Given that Omar just reaches the surface of the river before being pulled back up, find the value of \(L\) Fully justify your answer.
   8
2. If Omar is not modelled as a particle, explain the effect of revising this assumption on your answer to part (a).

8 In this question use \(g = 9.8 \mathrm {~m} \mathrm {~s} ^ { - 2 }\) A 'reverse' bungee jump consists of two identical elastic ropes. One end of each elastic rope is attached to either side of the top of a gorge. The other ends are both attached to Hannah, who has mass 84 kg
Hannah is modelled as a particle, as shown in the diagram below. \includegraphics[max width=\textwidth, alt={}, center]{f2470caa-0f73-4ec1-b08f-525c02ed2e67-12_467_844_678_598} The depth of the gorge is 50 metres and the width of the gorge is 40 metres.
Each elastic rope has natural length 30 metres and modulus of elasticity 3150 N
Hannah is released from rest at the centre of the bottom of the gorge.
8

1. Show that the speed of Hannah when the ropes become slack is \(30 \mathrm {~ms} ^ { - 1 }\) correct to two significant figures.
   8
2. Determine whether Hannah is moving up or down when the ropes become taut again. [5 marks] \includegraphics[max width=\textwidth, alt={}, center]{f2470caa-0f73-4ec1-b08f-525c02ed2e67-14_2492_1721_217_150} Additional page, if required.
   Write the question numbers in the left-hand margin. Question number Additional page, if required.
   Write the question numbers in the left-hand margin. Question number Additional page, if required.
   Write the question numbers in the left-hand margin. Question number Additional page, if required.
   Write the question numbers in the left-hand margin.

1 A bungee jumper of mass 80 kg steps off a high bridge with an elastic rope attached to her ankles. She is assumed to fall vertically from rest and the air resistance she experiences is modelled as a constant force of 32 N . The rope has natural length 4 m and modulus of elasticity 470 N . By considering energy, determine the total distance she falls before first coming to instantaneous rest.

A bungee jumper of weight \(W\) N is joined to a fixed point \(O\) by a light elastic rope of natural length \(20\) m and modulus of elasticity \(32\,000\) N. The jumper starts from rest at \(O\) and falls vertically. The jumper is modelled as a particle and air resistance is ignored.

1. Given that the jumper just reaches a point \(25\) m below \(O\), find the value of \(W\). [5]
2. Find the maximum speed reached by the jumper. [4]
3. Find the maximum value of the deceleration of the jumper during the downward motion. [3]

A bungee jumper of mass 80 kg steps off a high bridge with an elastic rope attached to her ankles. She is assumed to fall vertically from rest and the air resistance she experiences is modelled as a constant force of 32N. The rope has natural length 4 m and modulus of elasticity of 470 N. By considering energy, determine the total distance she falls before first coming to instantaneous rest. [6]