Kathy Kool buys a sports car that can accelerate at the rate of 5.00 m/s2. She decides to test the car by racing with another speedster, Stan Speedy. Both start from rest, but experienced Stan leaves the starting line 1.00 s before Kathy. Stan moves with a constant acceleration of 3.80 m/s2, and Kathy maintains an acceleration of 5.00 m/s2.
(a) Find the time it takes Kathy to overtake Stan from when she starts.
(b) Find the distance she travels before she catches him.
To solve this problem, we need to find the time it takes for Kathy to overtake Stan and the distance she travels before she catches him. We can use the kinematic equations of motion to find these values.
(a) Let's find the time it takes for Kathy to overtake Stan:
First, we'll find the time it takes for Stan to reach the point where Kathy starts. We'll use the equation:
d = v_i * t + (1/2) * a * t^2
Where:
d = distance traveled by Stan before Kathy starts (0 since they both start from rest)
v_i = initial velocity of Stan (0 since he starts from rest)
a = acceleration of Stan (3.80 m/s^2)
t = time
0 = 0 + (1/2) * 3.80 * t^2
Simplifying the equation above, we get:
1.9 * t^2 = 0
Since the only plausible solution for this equation is t = 0, we can conclude that Stan has already reached the starting line when Kathy starts.
Now, let's find the time it takes for Kathy to catch up to Stan. We'll use the following kinematic equation, where we consider Kathy as the object moving:
d = v_i * t + (1/2) * a * t^2
Where:
d = distance traveled by Kathy before catching Stan
v_i = initial velocity of Kathy (0 since she starts from rest)
a = acceleration of Kathy (5.00 m/s^2)
t = time
We also know that Kathy starts 1.00 second after Stan. So the equation becomes:
d = 0 * t + (1/2) * 5.00 * (t - 1)^2
Simplifying the equation above, we get:
d = 2.5 * (t - 1)^2
To find the time it takes for Kathy to catch up to Stan, we need to set the distance traveled by Kathy equal to the distance traveled by Stan when Kathy catches up. Therefore:
2.5 * (t - 1)^2 = 0 + (1/2) * 3.80 * t^2
Simplifying the equation above, we get:
2.5 * (t^2 - 2t + 1) = 1.9 * t^2
Expanding and rearranging terms, we have:
2.5t^2 - 5t + 2.5 = 1.9t^2
0.6t^2 - 5t + 2.5 = 0
Solving this quadratic equation will give us the time it takes for Kathy to catch up to Stan.