Problem

Two taxicab drivers decide to race their cabs. The first driver has a 30-second head start, and accelerates at 1 meter per second per second. The second driver accelerates at 2 meters per second per second. How many seconds will it take for the second driver to catch the first?

Solution

We'll call the distance the first driver travels d₁. Then we know from 2.1.4 in the textbook that the formula for d₁ is

The formula for the second driver is similar, but we must subtract 30 seconds from his time to account for the 30-second head start the first driver has.

To find when the second driver catches up to the first, we set these two distance formulas equal, and solve for t.

Use the quadratic formula or a calculator to factor the right hand side of the equation.

So solutions to the equation are t = 102.426 seconds and t = 17.574 seconds, approximately.

Which of these is correct?

When t = 17.574 seconds, the first driver has already been driving but the second driver hasn't started yet, so this cannot be the correct answer.

When t = 102.426 seconds, the two distance formulas give

Therefore, the second taxi will catch the first after 102.426 seconds.