Work Done by the Gravitational Force — Question 6

Question 6

The same gravity work need not produce the same final speed

Take a uniform gravitational field with $g=10$ m/s$^2$, upward y positive, and neglect air resistance. A 2 kg sled starts from rest and descends 3 m vertically along either of two fixed tracks. Track A is 6 m long; B is 10 m long. Each exerts a constant 2 N resisting force opposite motion. Normal forces do no work. Assume continuous descent is dynamically possible.

1. Find gravity work on each track.

2. Find total work and final speed on each.

3. Explain why equality of gravity work is insufficient to conclude equal final speeds.

Question 6 - Solution

The same gravity work need not produce the same final speed

(a) Use the height drop alone.

The governing relation is $$\boxed{W_{\mathrm{net}}=mg h-fL,\qquad v_f^2=2W_{\mathrm{net}}/m}$$

$$W_g=2(10)(3)=60\ \mathrm J$$ on either route.

(b) Include resistance over each path length.

A: $$W_R=-12\ \mathrm J,$$ net 48 J, so $$v_A=\sqrt{48}=4\sqrt3\ \mathrm{m/s}.$$ B: $$W_R=-20\ \mathrm J,$$ net 40 J, so $$v_B=\sqrt{40}=2\sqrt{10}\ \mathrm{m/s}.$$

(c) Compare the full force inventories.

The height fixes only gravity work. The longer route has more negative resistance work here, leaving less kinetic energy. It is net work, not gravity work alone, that determines the speed change.