A youngster bounces straight up and down on a trampoline. Suppose she doubles her initial speed from 2.0 m/s to 4.0 m/s.

(a) By what factor does her time in the air increase?

(b) By what factor does her maximum height increase?

(c) Verify your answer to part (a) with explicit calculations.

(d) Verify your answer to part (b) with explicit calculations.

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A 57kg trampoline artist jumps vertically upward from the top of a platform with a speed of 4.7m/s. How fast is he going as he lands on the trampoline 3.8m below?

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To answer these questions, we need to use some basic principles of physics related to motion and energy.

Let's start by understanding the concept of motion and energy when using a trampoline. When the youngster bounces up and down, her initial speed is the speed at the lowest point in her bounce, just before she starts moving back up. The maximum height is the highest point she reaches while bouncing. The time in the air is the total time it takes for her to complete one up-down cycle.

(a) To determine the factor by which her time in the air increases, we need to compare the time of the first bounce (initial speed of 2.0 m/s) with the time of the second bounce (initial speed of 4.0 m/s). Time is directly proportional to the height of the bounce, so we can use this relationship to find the factor of increase.

The formula we can use is:
(time in air) ∝ (height of bounce / initial speed)

Since we are comparing the case when the initial speed doubles, the height of the bounce remains the same for both cases. Therefore, the time in the air is directly proportional to the initial speed.

To find the factor of increase, we divide the time of the second bounce by the time of the first bounce:
(factor of increase) = (time in air with 4.0 m/s) / (time in air with 2.0 m/s)

We can derive the time in air using the formula for time of flight for an object in free fall. For simplicity, assuming ideal conditions without any air resistance, the time in air can be calculated as:
time = 2 * (initial velocity) / (acceleration due to gravity)

In this case, the acceleration due to gravity is the constant -9.8 m/s².

Let's calculate the time in air for both cases:
For initial speed of 2.0 m/s:
time1 = 2 * (2.0 m/s) / (-9.8 m/s²)

For initial speed of 4.0 m/s:
time2 = 2 * (4.0 m/s) / (-9.8 m/s²)

Now, we can calculate the factor of increase by dividing time2 by time1:
(factor of increase) = time2 / time1

(b) To determine the factor by which her maximum height increases, we can use the conservation of energy principle. The total mechanical energy of the system (the youngster and the trampoline) is conserved, which means the initial mechanical energy is equal to the maximum mechanical energy.

The mechanical energy of an object can be expressed as the sum of its kinetic energy (KE) and potential energy (PE):
Mechanical Energy (ME) = KE + PE

At the bottom of the bounce, all energy is in the form of kinetic energy, and at the top of the bounce, all energy is in the form of potential energy. Therefore, the mechanical energy at the bottom and at the top must be equal.

For simplicity, let's assume that the initial height and the maximum height are both measured from the ground level. Since the initial speed doubles from 2.0 m/s to 4.0 m/s, the kinetic energy at the bottom of the bounce doubles as well.

The kinetic energy (KE) is given by the formula:
KE = 0.5 * mass * (velocity)^2

Keeping in mind that mass is constant, the factor of increase in the maximum height will be equal to the factor of increase in the potential energy.

(c) To verify our answer for part (a) with explicit calculations, we substitute the values of time1 and time2 into the formula for (factor of increase) and calculate the result.

(factor of increase) = time2 / time1

(d) To verify our answer for part (b) with explicit calculations, we substitute the values of initial speed and maximum height into the formulas for kinetic energy and potential energy, and calculate the factor of increase in potential energy (which is also the factor of increase in maximum height).