How much work did the brakes do on a 900 kg automobile that increases in speed from 12.5 m/s to 14.6 m/s while descending a hill of 15.3 m?

To determine the work done by the brakes on the automobile, we can use the work-energy principle, which states that the work done on an object is equal to the change in its kinetic energy plus the change in gravitational potential energy.

First, we calculate the change in kinetic energy. The formula for kinetic energy (KE) is:

[

KE = \frac{1}{2}mv^2

]

where (m) is mass and (v) is velocity.

1. The initial kinetic energy when the car is traveling at 12.5 m/s is:

[

KE_{initial} = \frac{1}{2} \times 900 , \text{kg} \times (12.5 , \text{m/s})^2 = \frac{1}{2} \times 900 \times 156.25 = 70312.5 , \text{J}

]

2. The final kinetic energy when the car is traveling at 14.6 m/s is:

[

KE_{final} = \frac{1}{2} \times 900 , \text{kg} \times (14.6 , \text{m