Originally Posted by **meredith xo cheer** two cars are traveling along a straight road, one behind the other. the first is traveling at a constant velocity of 12 m/s. the second, approaching from the rear, is traveling at 25 m/s. when the second car is 200m behind the first, the driver applies the brakes, producing a constant acceleration of -0.20 m/s^2. will the cars crash and if so, where and when?
thansk so much to anyone that can help i really appreciate it! |

Let the origin be where the second car is at when the first applies the brakes. I am taking a positive direction as the direction of the initial velocity of either car.

We know that the second car's equation of motion is

$\displaystyle x = x_0 + vt$

$\displaystyle x = vt$

since the car starts at the origin.

For the first car we have

$\displaystyle x = x_0 + v_0t + \frac{1}{2}at^s$

where $\displaystyle x_0 = 200$, $\displaystyle v_0 = 12$, and $\displaystyle a = -0.2$ (in the appropriate units.)

The cars will meet at the same x values and the same t value. So we need to solve the system:

$\displaystyle x = 25t$

and

$\displaystyle x = 200 + 12t - 0.1t^2$

Subbing the first equation into the second:

$\displaystyle 25t = 200 + 12t - 0.1t^2$

$\displaystyle 0.1t^2 + 13r - 200 = 0$

Solve for t and sub it into either of the original equations to get x.

-Dan