Answer to Question #97853 in Mechanics | Relativity for Stefanie

Question

Consider the conservation of energy equation below.
ΔK + ΔU + ΔEint = W + Q + TMW + TMT + TET + TER
A ball of mass m falls from a height h to the floor.
(a) Identify the appropriate version of the equation above for the system of the ball and the Earth.
ΔEint = Q +TET + TER
or
ΔU = W
or
ΔK + ΔU = 0
or
ΔK = W
Use this equation to calculate the speed of the ball just before it strikes the Earth. (Use the following as necessary: m for mass, h for height, g for acceleration due to gravity.)
v =

(b) Identify the appropriate version of the equation above for the system of the ball.
ΔK = W
or
ΔEint = Q +TET + TER
or
ΔU = W
or
ΔK + ΔU = 0
Use this equation to calculate the speed of the ball just before it strikes the Earth. (Use the following as necessary: m for mass, h for height, g for acceleration due to gravity.)

Answer 1

(a) Since the ball falls from some height, it has potential energy, which turns into kinetic energy. Thus, the ball increases its kinetic energy by "consuming" its potential energy:

$$\Delta K+\Delta U=0,\\ \frac{1}{2}mv^2-mgh=0,\\ v=\sqrt{2gh}.$$

(b) The kinetic energy equals the work done by the gravitational field:

$$\Delta K=W,\\ \frac{1}{2}mv_f^2-\frac{1}{2}mv_i^2=mgh,\\ \frac{1}{2}mv_f^2-0=mgh,\\ v=\sqrt{2gh}.$$