Question

In Fig. 8-18, a horizontally moving block can take three frictionless routes, differing only in elevation, to reach the dashed finish line. Rank the routes according to (a) the speed of the block at the finish line and (b) the travel time of the block to the finish line, greatest first.

Short answer

a) The rank of routes according to the speed of the block at the finish line is 3,2,1

b) The rank of routes according to travel time of the block to the finish line, greatest first, is 1,2,3.

Step-by-step solution

Given information

A block moving horizontally along three frictionless roots

To understand the concept

By comparing the kinetic energy of the object throughout each path, we can rank the routes according to the speed of the block at the finish line. We can then rank the routes based on the travel time of the block to the finish line, as defined by the notion of velocity.

Formula:

The kinetic energy is given by,

$\mathrm{K}.\mathrm{E}.=\frac{1}{2}{\mathrm{Mv}}^2$

Velocity is given by,

$V=\frac{D}{T}$

(a) To rank the routes according to the speed of the block at the finish line.

For the object along the first route, it moves upward which is against the gravity, so gravity does work in the opposite direction on the object which reduces its kinetic energy. The kinetic energy is given by,

$\mathrm{K}.\mathrm{E}.=\frac{1}{2}{\mathrm{Mv}}^2$

Therefore, its speed slows down. Along the second route, the block moves without change in its speed. For the block along the third route, it moves in the direction of gravity. So, gravity does work in the same direction on the object which reduces its kinetic energy. Therefore, its speed increases.

Therefore, the ranking of routes according to the speed of the block at the finish line is 3,2,1.

(b) To rank the routes according to the travel time of the block to the finish line, greatest first.

The velocity in general is given by,

$v=\frac{D}{T}$

Therank of routes according to the speed of the block at the finish line is and the distance to travel is the same along all three routes. As speed is less, the time taken to travel a certain distance is more.

Therefore, the rank of routes according to the travel time of the block to the finish line, greatest first, is 1,2,3.