A drag racer, starting from rest, speeds up for 402 m with an acceleration of +24.0 m/s2. A parachute then opens, slowing the car down with an acceleration of -5.40 m/s2. How fast is the racer moving 3.75 102 m after the parachute opens

To find the speed of the drag racer 3.75 x 10^2 m after the parachute opens, we need to break down the problem into two parts:

1. Finding the speed of the drag racer after speeding up with an acceleration of +24.0 m/s^2 for a distance of 402 m.
2. Finding the speed of the drag racer after the parachute opens and it slows down with an acceleration of -5.40 m/s^2 for the remaining distance.

Let's calculate each part separately:

1. Finding the speed after speeding up:
To find the final speed (v) of the drag racer after covering a distance (d) with a constant acceleration (a), we can use the following kinematic equation:

v^2 = u^2 + 2ad

Where:
v = final velocity (unknown)
u = initial velocity (which is 0 m/s since the car starts from rest)
a = acceleration (+24.0 m/s^2)
d = distance (+402 m)

Rearranging the equation to solve for v:

v = √(u^2 + 2ad)

Substituting the given values into the equation:

v = √(0^2 + 2 * 24.0 * 402)
v = √(0 + 19344)
v ≈ √19344
v ≈ 139.08 m/s

So, the speed of the drag racer after speeding up is approximately 139.08 m/s.

2. Finding the speed after the parachute opens:
To find the speed after the parachute opens, we need to subtract the distance covered with the acceleration due to the parachute from the total distance.

The distance covered with the parachute's deceleration is:
distance_with_parachute = total_distance - distance_covered_while_speeding_up
distance_with_parachute = 3.75 x 10^2 m - 402 m
distance_with_parachute = 750 m - 402 m
distance_with_parachute ≈ 348 m

Now, using a similar equation as before, we can find the final speed (v) of the drag racer with the new acceleration (-5.40 m/s^2) over a given distance (distance_with_parachute).

v^2 = u^2 + 2ad

Where:
v = final velocity (unknown)
u = initial velocity (139.08 m/s)
a = acceleration (-5.40 m/s^2)
d = distance (348 m)

Rearranging the equation to solve for v:

v = √(u^2 + 2ad)

Substituting the given values into the equation:

v = √(139.08^2 + 2 * -5.40 * 348)
v = √(19317.9664 - 3764.16)
v ≈ √15553.8064
v ≈ 124.70 m/s

So, the speed of the drag racer 3.75 x 10^2 m after the parachute opens is approximately 124.70 m/s.