Question

c. on what time interval(s) do you think the bungee jumper achieves her greatest average velocity? why? d. estimate the jumper’s instantaneous velocity at t = 5. show your work

Explanation:

To answer this, we need to recall the concept of average velocity, which is defined as the change in position (displacement) divided by the change in time, i.e., \( v_{\text{avg}}=\frac{\Delta x}{\Delta t} \). For a bungee jumper, we typically analyze their motion: initially, when they jump, they are in free - fall (or a state of rapid downward motion) before the bungee cord starts to exert an upward force.

Step 1: Analyze the motion phases of a bungee jumper

• Free - fall phase (initial descent): When the bungee jumper first jumps off the platform, before the bungee cord becomes taut, she is accelerating downward due to gravity (neglecting air resistance for simplicity). In this phase, her displacement (change in position) is relatively large over a short period of time.
• Deceleration phase (after cord tautens): Once the bungee cord starts to stretch, the upward force from the cord increases. This upward force opposes the downward force of gravity. As the cord stretches more, the net force on the jumper decreases her downward acceleration and eventually can make her start to decelerate (slow down) or even start moving upward. In this phase, the rate of change of her position (displacement per unit time) will be less than in the initial free - fall phase.
• Oscillation phase: After reaching the lowest point, the bungee jumper will start to oscillate (move up and down) as the cord exerts a restoring force. In this phase, the average velocity over a time interval will be smaller because the jumper is moving both up and down, and the net displacement over a time interval will be relatively small.

Step 2: Determine the time interval for greatest average velocity

The greatest average velocity (in the downward direction, assuming we take downward as positive) occurs during the initial free - fall phase, that is, the time interval right after she jumps off the platform until the bungee cord becomes taut. This is because during this time:

• The displacement (\(\Delta x\)) is large (she is moving down a significant distance) as she is accelerating downward under gravity.
• The time interval (\(\Delta t\)) is relatively small. Since \( v_{\text{avg}}=\frac{\Delta x}{\Delta t} \), a large \(\Delta x\) and a small \(\Delta t\) will result in a large average velocity. In the later phases (when the cord is taut and during oscillation), either the displacement becomes smaller (because of deceleration or upward motion) or the time interval has to be considered over a period where the net displacement is not as large, leading to a smaller average velocity.

Answer:

The bungee jumper achieves her greatest average velocity during the time interval right after she jumps off the platform until the bungee cord becomes taut. This is because during this initial free - fall - like phase, she has a relatively large downward displacement (\(\Delta x\)) over a small time interval (\(\Delta t\)), and since average velocity \( v_{\text{avg}}=\frac{\Delta x}{\Delta t} \), a large \(\Delta x\) and small \(\Delta t\) give a large average velocity. In later phases (when the cord is taut or during oscillation), the rate of displacement (or net displacement over time) is smaller.