Chapter 8, Problem 1Q

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.

Figure 8-18 Question 1.

To determine

To find:

a) The ranking of routes according to the speed of the block at the finish line.

b) The ranking of routes according to the travel time of the block to the finish line, greatest first.

Answer to Problem 1Q

Solution:

a) The rank of routes according to the speed of the block at the finish line is $\mathrm{3,2},1.$

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

Explanation of Solution

1) Concept:

We can rank the routes according to the speed of the block at the finish line by comparing the kinetic energy of the object along them. And from this rank, according to the definition of velocity, we can rank the routes according to travel time of the block to the finish line.

2) Formula:

i)

$K.E.=\frac{1}{2}M{v}^2$

ii)

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

3) Given:

i) A block moving horizontally along three frictionless routes.

4) Calculation:

a) 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. We know that

$k.E=\frac{1}{2}M{v}^2$

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 $\mathrm{3,2},1.$

b) We have,

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

Therank of routes according to the speed of the block at the finish line is $\mathrm{3,2},1$ and 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 travel time of the block to the finish line, greatest first, is $\mathrm{1,2},3.$

Conclusion:

We can rank the routes according to the speed and travelling time of an object by comparing kinetic energy of the object along them.