1. Jane, looking for Tarzan, is running at top speed (4.9m/s) and grabs a vine hanging vertically from a tall tree in the jungle. How high can she swing upward?
2. In the high jump, Frank's kinetic energy is transformed into gravitational potential energy without the aid of a pole. With what minimum speed must Fran leave the ground in order to lift her center of mass 1.89m and cross the bar with a speed of 1.16m/s?
3. 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?
4. A 124g baseball is dropped from a tree 11.6m above the ground. With what speed would it hit the ground if air resistance could be ignored?
gain of of kinetic energy at low spot = loss of potential energy coming down from top
(m/2)(Vbottom^2 - Vtop^2) = m g (meters down from top)
or if stopped at top
(1/2)m v^2 = m g h
or (remember this) v = sqrt (2 g h)
To solve these questions, we will apply the principles of energy conservation and projectile motion. Let's break down each problem step by step:
1. Jane's maximum swing height:
To determine how high Jane can swing upward, we need to consider the conservation of mechanical energy. At the highest point of her swing, all of her initial kinetic energy will be converted into gravitational potential energy.
The initial kinetic energy (KE) can be calculated using the formula: KE = 0.5 * mass * velocity^2.
Since Jane's speed is given as 4.9 m/s and her mass is not provided, we cannot directly calculate the kinetic energy. To proceed, we will need additional information, such as her mass.
2. Frank's minimum speed in the high jump:
In this situation, Frank's kinetic energy is fully transformed into gravitational potential energy. The initial kinetic energy will be equal to the gravitational potential energy at the maximum height.
The formula for gravitational potential energy (PE) is: PE = mass * g * height, where g is the acceleration due to gravity.
In this case, the initial kinetic energy (KE) is equal to the gravitational potential energy (PE):
KE = PE
0.5 * mass * velocity^2 = mass * g * height
We are given that the height is 1.89m and the final speed is 1.16m/s. However, we don't have the value of gravity or the mass of Frank. As a result, we cannot directly calculate the minimum speed Frank needs to leave the ground. Additional information is required.
3. Trampoline artist's speed as he lands:
Similar to the previous question, we can solve this using energy conservation. At the highest point of his jump, all of the initial potential energy will be converted into kinetic energy.
The potential energy (PE) can be calculated using the formula: PE = mass * g * height.
Since the trampoline artist's mass is given as 57kg and the height is 3.8m, we can calculate the initial potential energy.
Next, we'll equate the initial potential energy to the final kinetic energy (KE) using the formula: KE = 0.5 * mass * velocity^2.
0.5 * mass * velocity^2 = mass * g * height
Simplifying the equation will allow us to solve for velocity.
4. Baseball hitting the ground speed:
In this problem, air resistance can be ignored. Therefore, the baseball's initial potential energy will be equal to its final kinetic energy at impact.
First, we calculate the initial potential energy (PE) using the formula: PE = mass * g * height.
Given a mass of 124g (0.124kg) and a height of 11.6m, we can determine the initial potential energy.
Next, we'll equate the initial potential energy to the final kinetic energy (KE) using the formula: KE = 0.5 * mass * velocity^2.
0.5 * mass * velocity^2 = mass * g * height
By simplifying the equation, we can solve for velocity.
Please note that obtaining the final answers for these questions requires additional data.