# A car moving with constant acceleration covered the distance between two points 52.4 m apart in 5.96 s. Its speed as it passes the second point was 14.6 m/s. (a) At what prior distance from the first point was the car at rest?

##### 1 Answer

The car was at rest

#### Explanation:

The idea here is that once the car **starts moving**, its acceleration will be *constant*.

This means that you can use the speed with which it passes the first point and its acceleration to write

#v_1^2 = underbrace(v_0^2)_(color(blue)(=0)) + 2 * a * d" "# , where

**at the moment** it passes the first point;

**before** it passes the first point.

The values you have for

You basically work with two equations with two unknowns,

First, you know that

#s = v_1 * t + 1/2 * a * t^2" "# , where

SInce you know that car's speed when it reaches the second point, you can write

#v_2 = v_1 + a * t implies v_1 = v_2 - a * t#

Plug this back into the above equation to get

#s = (v_2 - at) * t + 1/2 * a * t^2#

Plug in your values to get - I'll skip the units for simplicity

#52.4 = (14.6 - 5.96 * a) * 5.96 + 1/2 * a * 5.96""^2#

This will eventually get you

#52.4 = 87.016 - 35.52a + 17.76a#

#17.76a = 34.616 implies a = 34.616/17.76 = "1.95 m/s"""^2#

The speed of the car **at the time it passes the first point** will be

#v_1 = 14.6 - 1.95 * 5.96 = "2.99 m/s"#

Now, you know that the car started from rest and reached a speed of

#v_1^2 = 2 * a * d implies d = v_1^2/(2 * a) = (2.99^2"m"^color(red)(cancel(color(black)(2)))/color(red)(cancel(color(black)("s"^2))))/(2 * 1.95color(red)(cancel(color(black)("m")))/color(red)(cancel(color(black)("s"^2)))) = color(green)("2.29 m")#

Alternatively, you can first find the time it took for the car to reach the first position

#v_1 = underbrace(v_0)_(color(blue)(=0)) + a * t_b implies t_b = v_1/a = (2.99color(red)(cancel(color(black)("m")))/color(red)(cancel(color(black)("s"))))/(1.95color(red)(cancel(color(black)("m")))/"s"^color(red)(cancel(color(black)(2)))) = "1.53 s"#

The distance it covered before the first position is

#d = 1/2 * a * t_b^2 = 1/2 * 1.95"m"/color(red)(cancel(color(black)("s"^2))) * 1.53""^2color(red)(cancel(color(black)("s"^2))) = color(green)("2.29 m")#