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
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")#