Question

Claire starts driving with an initial velocity of 5.0 m/s towards north. She drives along a straight, level path with a uniform acceleration and travels 500. m in 20. seconds.

A. What is the average velocity for the entire 20. seconds?

B. What is the velocity at the end of 20. seconds?

Answer 1

When you want to calculate AVERAGE velocity in the presence of constant acceleration, you can safely use v=Δd/Δt to calculate your answer. In this case that will be 500/20=25 m/s.

The question gives us initial velocity, distance, and time. To calculate the final velocity at 20 s, we can put together a couple kinematics equations:

\[(1) \space \space v_f^2=v_i^2+2aΔ d\]\[(2) \space \space a = \frac{Δv}{Δt}\]
Sub 2 back into 1 so that way we get rid of acceleration and only work with variables we know the value of to solve for final velocity:
\[v_f^2=v_i^2+2 \frac{Δv}{Δt}Δd\]
Note that Δv is final velocity minus initial velocity, Δt is just t, and Δd is just d. So:
\[v_f^2=v_i^2+2 \frac{v_f-v_i}{t}d\]\[v_f^2t=v_i^2t+2dv_f-2dv_i\]
I know this looks gross, but we can fix that - let's plug in the numbers we know:
\[20v_f^2=500+1000v_f-5000\]
Now put this in standard from and divide everything by 20 just to make things a bit easier:
\[v_f^2-50v_f+225=0\]
Solve this quadratic and you'll find that v=5 and v=45. We know that we're accelerating uniformly from 5 m/s, so our final speed can't also be 5 m/s. That means we can reject that value of v and go with 45 m/s as our final velocity!

Whew!

Let me know if you have any questions!