# A car traveling 58.5 km/h is 27.4 m from a barrier when the driver slams on the brakes. The car hits the barrier 2.18 s later. (a) What is the car's constant deceleration magnitude before impact? (b) How fast is the car traveling at impact ?

Sep 2, 2015

$a = - {\text{3.38 m/s}}^{2}$
${v}_{\text{impact" = "8.88 m/s}}$

#### Explanation:

Start by converting the car's initial velocity from kilometers per hour to meters per second

58.5color(red)(cancel(color(black)("km")))/color(red)(cancel(color(black)("h"))) * (1color(red)(cancel(color(black)("h"))))/(60color(red)(cancel(color(black)("min")))) * (1color(red)(cancel(color(black)("min"))))/(60color(red)(cancel(color(black)("s")))) * "1000 m"/(1color(red)(cancel(color(black)("km")))) = "16.25 m/s"

Since the car doesn't change its direction of movement, you can say that its velocity will be equal to its speed.

So, you know that the car's velocity when it begins to break is equal to 16.25 m/s. Moreover, you know that it took the car 2.18 seconds to cover a distance of 27.4 meters with constant deceleration.

This means that you can write

$\textcolor{b l u e}{d = {v}_{0} \cdot t + \frac{1}{2} \cdot a \cdot {t}^{2}} \text{ }$, where

$d$ - the distance to the barrier;
${v}_{0}$ - the initial speed of the car;
$t$ - the stopping time;
$a$ - the deceleration - you can expect it to be negative;

You can expect the acceleration to be negative because if you take the direction of movement to be positive, then for a car that is slowing down, its acceleration vector will be oriented in the opposite direction of its motion.

Rearrange this equation to solve for $a$

$d = {v}_{0} \cdot t = \frac{1}{2} \cdot a \cdot {t}^{2}$

$a = \frac{2 \cdot \left(d - {v}_{0} \cdot t\right)}{t} ^ 2$

a = (2 * (27.4"m" - 16.25"m"/color(red)(cancel(color(black)("s"))) * 2.18color(red)(cancel(color(black)("s")))))/(2.18^2 "s"""^2) = color(green)(-"3.38 m/s"""^2

To determine the car's speed upon impact, use the equation

color(blue)(v_"impact" = v_0 + a * t)" ", where

${v}_{\text{impact}}$ - the speed upon impact;

Plug in your values to get

v_"impact" = 16.25"m"/"s" + (-3.38)"m"/"s"^color(red)(cancel(color(black)(2))) * 2.18color(red)(cancel(color(black)("s"))) = color(green)("8.88 m/s")