Answer to Question #116745 in Classical Mechanics for NILAKSHA GHOSH

Question

A 1kg mass slides down several frictionless inclines - each incline angle decreases by 1 degree for each 1-meter segment travelled. When the mass reaches the 0-degree incline, determine the total time travel time, the total height the mass falls and velocity.

Answer 1

If the first meter inclined 90°, the other 89°, and so on, the total height will be:

$$H=d\text{ sin}90^\circ+d\text{ sin}89^\circ+...+d\text{ sin}23^\circ+...+d\text{ sin}1^\circ= \\=d\sum^{90^\circ}_{i=1}\text{sin}n_i=57.79\text{ m}.$$

Using energy conservation, the final velocity:

$$v=\sqrt{2gH}=\sqrt{2\cdot9.8\cdot57.79}=33.7\text{ m/s}.$$

Ignoring friction and jumps at the edges where inclines are joined, we can try to calculate the time if that mass is falling from height H:

$$t=\sqrt{\frac{2H}{g}}=\sqrt{\frac{2\cdot57.79}{9.8}}=3.43\text{ s},$$

and compare it with real time calculated according to the expression

$$t=\sum^{90}_{i=1}t_i,\\ \space\\ t_i=\frac{v_i+v_{i-1}}{2d}=\frac{v_i+v_{i-1}}{2d},\\ \space\\ v_i=\sqrt{2gd\text{ sin}(90-i)^\circ}.\\ t=33.7\text{ s}.$$