Question

A 10 kg ball falls form the height of 100 m (a) calculate the vertical speed of the ball during the first 4 seconds ,(b) calculate the height of the ball above the gorund during the first 4 seconds ,(c) calculate the kinetic and potential energies of the ball.(d) determine the total mechanical energy of the ball?

Answer 1

Let positive y-axis be pointing up. Then

$$v(t)=v_0-gt=-gt, v_0=0$$

$$y(t)=y_0+v_0t-\dfrac{gt^2}{2}=y_0-\dfrac{gt^2}{2}, v_0=0, y_0=100\ m$$

(a)

$$v(4)=-9.81\ m/s^2(4\ s)=-39.24\ m/s$$

(b)

$$y(4)=100\ m-\dfrac{9.81\ m/s^2(4\ s)^2}{2}=21.52\ m$$

(c)

$$P=mgy$$

$$P|_{t=0}=10\ kg(9.81\ m/s^2)(100\ m)=9810\ J$$

$$P|_{t=4}=10\ kg(9.81\ m/s^2)(21.52\ m)=2111.112\ J$$

$$K=\dfrac{mv^2}{2}$$

$$K|_{t=0}=0$$

$$K|_{t=4}=\dfrac{10\ kg(-39.24\ m/s)^2}{2}=7698.888\ J$$

(d)

$$E=P+K=const$$

$$E=9810\ J+0\ J=9810\ J$$

$$=2111.112\ J+7698.888\ J$$

The total mechanical energy of the ball is 9810 J and is conserved during the motion.