A young woman named Kathy Kool buys a
sports car that can accelerate at the rate of
5.14 m/s
2
. She decides to test the car by drag
racing with another speedster, Stan Speedy.
Both start from rest, but experienced Stan
leaves the starting line 0.78 s before Kathy.
Stan moves with a constant acceleration of
3.68 m/s
2
and Kathy maintains an acceleration of 5.14 m/s
2
.
Find the time it takes Kathy to overtake
Stan.
Answer in units of s
To find the time it takes Kathy to overtake Stan, we need to determine when their positions are equal.
Let's first calculate the position of Stan after 0.78 seconds using the equation of motion:
For Stan:
Initial velocity (u) = 0 (since he starts from rest)
Acceleration (a) = 3.68 m/s^2
Time (t) = 0.78 s
Using the equation:
s = ut + (1/2)at^2
Substituting the values:
s = 0 * 0.78 + (1/2) * 3.68 * (0.78)^2
Calculating, we find:
s = 0 + (1/2) * 3.68 * 0.6084
s ≈ 0.713 m
So, Stan's position after 0.78 seconds is approximately 0.713 meters.
Now, let's find the time it takes Kathy to overtake Stan. We need to determine the time for which their positions are equal.
For Kathy:
Initial velocity (u) = 0 (since she starts from rest)
Acceleration (a) = 5.14 m/s^2
Time (t) = t (unknown)
Using the equation:
s = ut + (1/2)at^2
Substituting the values, with s being the distance traveled by Kathy when she overtakes Stan (equal to 0.713 m):
0.713 = 0 * t + (1/2) * 5.14 * t^2
Rearranging and simplifying the equation:
2.57t^2 = 0.713
Dividing both sides by 2.57:
t^2 = 0.277
Taking the square root on both sides to solve for t:
t ≈ √0.277
t ≈ 0.526 s
Therefore, it takes Kathy approximately 0.526 seconds to overtake Stan.