speed - distance catching up

*A speeding motorbike travels past a stationary police car. The police car starts accelerating immediately, and keeps accelerating until it has passed the bike.*

DATA: motorbike speed: $\displaystyle 35ms^{-1}$; police car acceleration: $\displaystyle 4ms^{-2}$

How far does the police car travel before it overtakes the motorbike?

I tried working this out using the following method, but it is wrong. How come?

**My Solution**

Let the distance travelled by the motorbike be $\displaystyle d_m$ and the distance travelled by the police car be $\displaystyle d_p$. Let $\displaystyle t$ be time in seconds.

The distance travelled by the motorbike is given by

$\displaystyle d_m=35t$

The distance travelled by the police car can be expressed by

$\displaystyle d_p=4+8+...+t=\sum_{i=1}^t 4i = \frac{4t(t+1)}{2}=2t(t+1)$

When $\displaystyle d_p=d_m$, $\displaystyle 35t=2t(t+1)$

Solving gives $\displaystyle t=16.5 \Rightarrow d = 577.5 \mbox{m}$

Thankyou!