Question

The acceleration of a car t seconds after the driver steps on the brake, before the car comes to a full stop, is a(t)=?(t?5)2 mph per second. What's the decrease in velocity in mph 4 seconds after the brake is applied?

Answer 1

The decrease in velocity of the car 4 seconds after the brake is applied is 124/3 mph per second

To find the decrease in velocity of the car 4 seconds after the brake is applied, we need to integrate the acceleration function from 0 to 4. The given acceleration function is a(t) = (t - 5)^2 mph per second.

To find the change in velocity, we integrate the acceleration function with respect to time:

∫[(t - 5)^2] dt

Using the power rule of integration, we can expand and integrate the function:

= ∫[t^2 - 10t + 25] dt

= (t^3/3 - 10t^2/2 + 25t) + C

Now, we evaluate the integral from 0 to 4:

Change in velocity = [(4^3/3 - 10(4)^2/2 + 25(4)) - (0^3/3 - 10(0)^2/2 + 25(0))]

= [(64/3 - 80/2 + 100) - (0 - 0 + 0)]

= [(64/3 - 40 + 100) - (0)]

= 124/3 mph per second

Therefore, the decrease in velocity of the car 4 seconds after the brake is applied is 124/3 mph per second.