Question

A bungee jumper with mass \(55.0 \mathrm{~kg}\) reaches a speed of \(13.3 \mathrm{~m} / \mathrm{s}\) moving straight down when the elastic cord tied to her feet starts pulling her back up. After \(1.25 \mathrm{~s}\), the jumper is heading back up at a speed of \(10.5 \mathrm{~m} / \mathrm{s} .\) What is the average force that the bungee cord exerts on the jumper? What is the average number of \(g^{\prime}\) 's that the jumper experiences during this direction change?

Short answer

Solution: 1. Calculate the change in momentum: Initial momentum: \(p_i = m v_i = 55.0\ \mathrm{kg} × 13.3\ \mathrm{m/s} = 731.5\ \mathrm{kg\ m/s}\) Final momentum: \(p_f = m v_f = 55.0\ \mathrm{kg} × -10.5\ \mathrm{m/s} = -577.5\ \mathrm{kg\ m/s}\) Change in momentum: \(Δp = p_f - p_i = -577.5\ \mathrm{kg\ m/s} - 731.5\ \mathrm{kg\ m/s} = -1309\ \mathrm{kg\ m/s}\) 2. The total time taken is given: 1.25 seconds. 3. Calculate the average force exerted by the bungee cord: Average Force: \(F_{avg} = \frac{Δp}{t} = \frac{-1309\ \mathrm{kg\ m/s}}{1.25\ \mathrm{s}} = -1047.2\ \mathrm{N}\) 4. Calculate the average acceleration: Total change in velocity: \(Δv = v_f - v_i = -10.5\ \mathrm{m/s} - 13.3\ \mathrm{m/s} = -23.8\ \mathrm{m/s}\) Average acceleration: \(a_{avg} = \frac{Δv}{t} = \frac{-23.8\ \mathrm{m/s}}{1.25\ \mathrm{s}} = -19.04\ \mathrm{m/s^2}\) 5. Convert average acceleration to the number of g's experienced: Number of g's: \(\frac{a_{avg}}{g} = \frac{-19.04\ \mathrm{m/s^2}}{9.81\ \mathrm{m/s^2}} = -1.94\ \mathrm{g}\) Answer: The average force exerted by the bungee cord on the jumper is -1047.2 N, and the average number of g's experienced during the direction change is -1.94 g.

Step-by-step solution

Calculate the change in momentum

We will first find the initial downward momentum and final upward momentum. The change in momentum can be found by subtracting the initial momentum from the final momentum. The formula for momentum (p) is given by: p = mv where m is the mass of the jumper, and v is the velocity. Initial momentum: \(p_i = m v_i\) where \(m=55.0\ \mathrm{kg}\) and \(v_i=13.3\ \mathrm{m/s}\) (downward) Final momentum: \(p_f = m v_f\) where \(v_f=-10.5\ \mathrm{m/s}\) (upward) Change in momentum: \(Δp = p_f - p_i\)

Determine the total time taken for this change

The problem states that it takes 1.25 seconds for the jumper's velocity to change to upward. This is the total time taken for the change in momentum. Total time: \(t = 1.25\ \mathrm{s}\)

Calculate the average force exerted by the bungee cord

Now that we have determined the change in momentum and the total time taken we can calculate the average force using the equation: Average Force: \(F_{avg} = \frac{Δp}{t}\)

Calculate the average acceleration

The average acceleration can be found by dividing the total change in velocity by the total time taken: Average acceleration: \(a_{avg} = \frac{Δv}{t}\)

Convert average acceleration to the number of g's experienced

To find the average number of g's experienced, we can simply divide the average acceleration by the acceleration due to gravity. Number of g's: \(\frac{a_{avg}}{g}\) where \(g = 9.81\ \mathrm{m/s^2}\) Following these steps, we can find the average force exerted by the bungee cord and the average number of g's experienced by the jumper during the direction change.

Key concepts

Momentum and Its Conservation in Bungee Jumping

Momentum, a measure of the motion of a body, is the product of its mass and velocity, expressed by the equation \( p = mv \). A bungee jumper demonstrates the principle of conservation of momentum — the total momentum before and after jumping is constant when no external forces act aside from gravity and the bungee cord.

When the jumper reaches the end of their free fall and the cord stretches, their velocity and thereby momentum change direction. This change in momentum, calculated by finding the difference between the final and initial momentum as \( \Delta p = p_f - p_i \), is what the bungee cord needs to counteract to stop the fall and initiate the rebound. Since momentum is a vector quantity, it's crucial to take into account the direction, which is why the change includes a sign reversal.

Calculating Average Force in Bungee Jumping Dynamics

The average force exerted by the bungee cord can be derived from the rate of change of momentum. By dividing the change in momentum by the total time over which this change occurs, we obtain the average force as \( F_{avg} = \frac{\Delta p}{t} \).

The average force is an essential factor in designing bungee cords, as it determines the forces experienced by both the cord and the jumper, which must be within safe limits. In the case of our bungee jumper, the average force provides a glimpse into the impact of the bungee cord's tension and its capability to safely decelerate and then accelerate the jumper in the opposite direction.

Acceleration During the Bungee Jump

Acceleration is the rate of change of velocity with respect to time. In a bungee jump, acceleration can vary throughout the jump due to the uncoiling and recoiling of the cord, which adds complexity beyond the constant acceleration of gravity experienced during free fall. This is where average acceleration comes into play.

The average acceleration is particularly helpful for understanding the overall effect of the bungee cord on the jumper. It is calculated by the total change in velocity, considering both magnitude and direction, divided by the time this change takes, as shown by \( a_{avg} = \frac{\Delta v}{t} \). It tells us how quickly the bungee cord can alter the direction and speed of the jumper.

Understanding Change in Velocity During Bungee Jumps

Change in velocity not only signifies a change in speed but also a change in direction. In bungee jumping, 'change in velocity' is apparent when the jumper goes from falling down to moving upwards.

Calculating this change involves subtracting the initial downward velocity from the final upward velocity. Since direction matters, the final velocity is taken as negative when the jumper is moving back up. This change, when combined with the mass and the time duration, allows us to understand and calculate the average force via the change in momentum and further permits us to compute the average number of g's the jumper experiences. The concept of 'change in velocity' is therefore fundamental to analyzing the motions involved in bungee jumping.