# 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 ?

##### 1 Answer

#### Answer:

#### 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

#color(blue)(d = v_0 * t + 1/2 * a * t^2)" "# , where

#d# - the distance to the barrier;

#v_0# - the initial speed of the car;

#t# - the stopping time;

#a# - thedeceleration- you can expect it to benegative;

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

#d = v_0 * t = 1/2 * a * t^2#

#a = (2 * (d - v_0 * t))/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

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")#