Exercise 7.10
A ball of mass kg is thrown up with a velocity of m . (i) Identify the sign of the work done by gravity on the ball during its upward motion and its downward motion. (ii) If the ball reaches a height of m, how much work was done by air resistance (assume g = m ).
Disagrees with the book
This working does not reach the answer printed in NCERT. One of the two is wrong and it has not yet been settled which — check it before you rely on it.
NCERT’s answer
(i)
Negative and positive (ii) - $\displaystyle 12$ J
(i) Upward motion: work done by gravity is negative. Downward motion: it is positive.
Going up, gravity acts downward while the displacement is upward — opposite directions, so negative work.
Coming down, gravity and the displacement are both downward — same direction, so positive work.
(ii) Work done by air resistance = \(\displaystyle -12\ \text{J} \).
Initial kinetic energy: \(\displaystyle K_{i} = \frac{1}{2}mv^{2} = \frac{1}{2} \times 2\ \text{kg} \times (20\ \text{m s}^{-1})^{2} = 400\ \text{J} \)
At the top the ball is momentarily at rest, so \(\displaystyle K_{f} = 0\ \text{J} \).
Total work done on the ball \(\displaystyle = K_{f} - K_{i} = 0 - 400 = -400\ \text{J} \)
Work done by gravity over the $\displaystyle 19.4$ m rise \(\displaystyle = -mgh = -(2\ \text{kg} \times 10\ \text{m s}^{-2} \times 19.4\ \text{m}) = -388\ \text{J} \)
The only other force is air resistance, so \(\displaystyle W_{\text{air}} = -400\ \text{J} - (-388\ \text{J}) = -12\ \text{J} \)
It is negative because air resistance always opposes the motion. Without it the ball would have risen to $\displaystyle 20$ m.