how fast must a ball be thrown upward to reach a height of 12 m?
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How fast must a ball be thrown upward to reach a height of 12m?
To find out how fast a ball must be thrown upward to reach a height of 12 m, you can use the principles of projectile motion and conservation of energy. Here's a step-by-step explanation of how to solve this:
1. Start by using the conservation of energy equation, which states that the total mechanical energy of an object remains constant throughout its motion. In this case, the ball's mechanical energy consists of its kinetic energy (KE) and potential energy (PE):
KE + PE = constant
2. Since the ball is thrown upward, we assume the initial velocity is positive and the final velocity (at the peak of its trajectory) will be zero. This allows us to focus on the changes in potential energy (ΔPE) during its motion.
ΔPE = PE_final - PE_initial
3. At the peak of the ball's trajectory, the height is maximum, so the potential energy is at its highest point. Thus, at the top of its flight, PE_final = mgh, where m is the mass of the ball, g is the acceleration due to gravity (approximately 9.8 m/s²), and h is the height.
PE_final = mgh
4. At the start of the ball's motion, it has kinetic energy but no potential energy. Therefore, PE_initial = 0.
PE_initial = 0
5. Applying the conservation of energy equation, we have:
ΔPE = PE_final - PE_initial = mgh - 0 = mgh
6. The change in potential energy (ΔPE) is equal to the negative change in kinetic energy (-ΔKE) since the ball slows down as it rises. ΔKE is given by:
ΔKE = -ΔPE = -mgh
7. The change in kinetic energy is equal to the initial kinetic energy (KE_initial) minus the final kinetic energy (KE_final) since the velocity at the top of the trajectory is zero. KE_final = 0.
ΔKE = KE_final - KE_initial = 0 - KE_initial = -mgh
8. Solving for the initial kinetic energy, we have:
KE_initial = mgh
9. At the start of the motion, the ball has only kinetic energy. Therefore, the initial kinetic energy is equal to 1/2 times the mass of the ball times the square of its initial velocity.
KE_initial = 1/2 mv²
10. Equating the expressions for initial kinetic energy:
1/2 mv² = mgh
11. Simplifying the equation:
v² = 2gh
12. Finally, to find the initial velocity (v), take the square root of both sides:
v = √(2gh)
By plugging in the values of g (acceleration due to gravity) and h (height), you can calculate the required initial velocity to achieve a height of 12 m.