You are thinking about taking gymnastics, so you go to the facility and get an idea of what to expect by looking out from the viewing room. The viewing room window is 4.30 m above the trampoline directly below, so it is perfect for viewing the the facility. Occasionally someone jumps past the window and then back down. On one occurrence a gymnast went up past the window and came back down; as he passed the window on the way down, you notice that his speed is 14.1 m/s.
What must have been his initial speed coming off the trampoline?
speed down equals speed up
1/2 m Vo^2 = 1/2 m v^2 + m g h
Vo^2 = v^2 + 2 g h = 14.1^2 + (2 * 9.80 * 4.30)
that is SOME trampoline ...
To determine the initial speed of the gymnast coming off the trampoline, we can use the concepts of conservation of energy and projectile motion. Here's how you can find the answer:
Step 1: Understand the concept
When the gymnast jumps on a trampoline, they convert potential energy into kinetic energy. The potential energy at the highest point of the jump is equal to the kinetic energy just before hitting the trampoline again.
Step 2: Calculate the potential energy
The potential energy is given by the formula: Potential Energy = mass * gravitational acceleration * height. Here, the mass of the gymnast is not given, but it cancels out when we compare the potential and kinetic energies, so it is not needed.
Step 3: Substitute the values
In this case, the height is the distance from the trampoline to the window, which is 4.30 m. The gravitational acceleration is approximately 9.8 m/s^2. Plugging these values into the formula, the potential energy is 4.30 m * 9.8 m/s^2 = 42.14 Joules.
Step 4: Apply conservation of energy
The total mechanical energy of the gymnast (sum of kinetic and potential energies) remains constant throughout the jump. Since we know the kinetic energy just before hitting the trampoline (because it is given), we can equate it to the potential energy at the highest point.
Step 5: Calculate the initial speed
The kinetic energy is given by the formula: Kinetic Energy = 0.5 * mass * (initial speed)^2. Rearranging the formula, we can solve for the initial speed: (initial speed)^2 = 2 * Kinetic Energy / mass. Again, we don't know the mass of the gymnast, but it cancels out.
Step 6: Substitute the values
The given kinetic energy is 0.5 * mass * (14.1 m/s)^2. Plugging in this value, we get: (initial speed)^2 = 2 * (0.5 * mass * (14.1 m/s)^2). Since the mass cancels out, the equation becomes: (initial speed)^2 = (14.1 m/s)^2.
Step 7: Solve for initial speed
Taking the square root of both sides of the equation, we find: initial speed = 14.1 m/s.
Therefore, the initial speed of the gymnast coming off the trampoline must have also been 14.1 m/s.