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 solve this problem, you can break it down into two parts: the initial acceleration phase and the deceleration phase.
First, let's calculate the final velocity during the initial acceleration phase. We can use the kinematic equation:
v^2 = u^2 + 2as
Where:
- v is the final velocity
- u is the initial velocity (which is 0 m/s since the car starts from rest)
- a is the acceleration during the initial phase (in this case, +24.0 m/s^2)
- s is the displacement (402 m)
Plugging in the values, the equation becomes:
v^2 = 0 + 2 * 24.0 * 402
Simplifying:
v^2 = 19,296
Taking the square root of both sides:
v = √(19,296)
v ≈ 138.9 m/s (rounded to one decimal place)
Now, for the deceleration phase, we need to calculate the time it takes for the car to travel 3.75 * 10^2 m. To do this, we can use the equation:
s = ut + (1/2)at^2
Where:
- s is the displacement (3.75 * 10^2 m)
- u is the initial velocity during the deceleration phase (which is the final velocity from the previous step since it continues from there)
- a is the acceleration during the deceleration phase (in this case, -5.40 m/s^2)
- t is the time
Rearranging the equation, we get:
t^2 - [(2s) / a] = 0
Plugging in the values, the equation becomes:
t^2 - [(2 * 3.75 * 10^2) / -5.40] = 0
Simplifying:
t^2 ≈ 138.9
Taking the square root of both sides:
t ≈ √(138.9)
t ≈ 11.8 s (rounded to one decimal place)
Finally, we can calculate the final velocity during the deceleration phase using the equation:
v = u + at
Where:
- v is the final velocity during the deceleration phase
- u is the initial velocity during the deceleration phase (which is the final velocity from the previous step)
- a is the acceleration during the deceleration phase (in this case, -5.40 m/s^2)
- t is the time (11.8 s)
Plugging in the values, the equation becomes:
v = 138.9 + (-5.40) * 11.8
Simplifying:
v ≈ 77.7 m/s (rounded to one decimal place)
Therefore, the racer is moving at approximately 77.7 m/s 3.75 * 10^2 m after the parachute opens.