Problem 81 - University Physics with Modern Physics

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 \(0.0250 \mathrm{~s},\) 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 \(g\) 's that the jumper experiences during this direction change?

Step-by-Step Solution

Answer

Answer: ____________ N, ____________ g's

Step 1: Determine initial and final momentum

Calculate the momentum before and after the elastic cord starts pulling. Momentum (p) is given by the product of mass (m) and velocity (v). Initial momentum (\(p_1\)): \(p_1 = m * v_1\), with \(m = 55.0 kg\) and \(v_1 = -13.3 m/s\). Final momentum (\(p_2\)): \(p_2 = m * v_2\), with \(m = 55.0 kg\) and \(v_2 = 10.5 m/s\).

Step 2: Calculate the impulse

The impulse (J) on the jumper can be found by the change in momentum \(\Delta p\): \(J = \Delta p = p_2 - p_1\). Calculate the impulse on the jumper by substituting the values.

Step 3: Calculate the average force

The impulse-momentum theorem states that the impulse is equal to the average force (F) multiplied by the time interval \(\Delta t\). Therefore, \(J = F * \Delta t\) We can now solve for the average force acting on the jumper: \(F = \frac{J}{\Delta t}\), where \(\Delta t = 0.0250 s\).

Step 4: Determine the average number of g's

To find the average number of g's experienced by the jumper, we need to divide the average force by the jumper's weight: Average number of g's = \(\frac{F}{mg}\), where \(g \approx 9.81 m/s^2\). Calculate the value using the average force found in Step 3.

Key Concepts

Momentum

Momentum is a physical quantity that represents the motion contained within an object. It is directly related to the object's mass and velocity, described with the equation,
\( p = m \times v \),
where \( p \) is the momentum, \( m \) is the mass, and \( v \) is the velocity. In the context of bungee jumping, as the jumper hurtles downwards, the mass remains constant, but the velocity changes notably when the cord begins to stretch and decelerate the jumper. This change causes a shift in momentum, which is pivotal to understanding the forces at play during the jump.

Impulse

Impulse is defined as the change in momentum of an object when a force is applied over a period of time. It is given by,
\( J = m \times \Delta v \),
where \( J \) is the impulse, \( m \) is the mass, and \( \Delta v \) is the change in velocity. For the bungee jumper, the impulse would be the product of their mass and the difference between their final and initial velocity. This concept explains the effectiveness of the bungee cord in altering the jumper's speed in such a short interval.

Impulse-Momentum Theorem

The impulse-momentum theorem is integral to the physics of bungee jumping. It states that the impulse on an object is equal to the change in its momentum. This can be expressed with the equation,
\( J = \Delta p \),
where \( J \) is the impulse and \( \Delta p \) is the change in momentum. By using the initial and final velocity of our bungee jumper, and knowing their mass, we can calculate the impulse required to change their direction. This impulse is provided by the tension in the bungee cord, which can then be used to find out the force exerted on the jumper.

Average Force

Average force is the total force executed on an object divided by the time it was applied. It's especially useful to find the average force exerted by the bungee cord as it brings the jumper to a stop and then propels them upwards. According to the impulse-momentum theorem, you can calculate the average force using the following relationship,
\( F = \frac{J}{\Delta t} \),
where \( F \) is the average force, \( J \) is the impulse, and \( \Delta t \) is the time over which the impulse is applied. By deciphering the average force, we estimate the immense stresses a bungee jumper's body undergoes in the brief moment of direction change.

Acceleration Due to Gravity

The acceleration due to gravity, denoted as \( g \), is approximately \( 9.81 \frac{m}{s^2} \) on Earth's surface. It's a constant acceleration that acts on objects in freefall, governing their increase in speed as they descend. For a bungee jumper, this acceleration plays a crucial role until the cord begins to stretch. When analyzing the force felt by the jumper, we consider it in relation to their weight, a product of mass and gravity, to express it in g-forces. These g-forces are a measure of the acceleration and provide insight into the physical experience of the jumper.