A young woman named Kathy Kool buys a sports car that can accelerate at the rate of 4.53 m/s2. She decides to test the car by dragging with another speedster, Stan Speedy. Both start from rest, but experienced Stan leaves the starting line 1.01 s before Kathy. If Stan moves with a constant acceleration of 3.62 m/s2 and Kathy maintains an acceleration of 4.53 m/s2,
(A)Find the time it takes Kathy to overtake Stan.
(B) Calculate the distance she travels before she catches him.
(C) Calculate the speed of Kathy's car at the instant she overtakes Stan.
(D) Calculate the speed of Stan's car at the instant he is overtaken by Kathy.
d1 = 0.5a*t^2 = 1.81*1.01^2 = 1.85 m. Head start.
V1 = a*t = 3.62*1.01 = 3.66 m/s.
d2 = 0.5a*t^2 = 1.85 + V1*t + 0.5a*t^2
2.27t^2 = 1.85 + 3.66*t + 1.81t^2
2.27t^2-1.81t^2 -3.66t - 1.85 = 0
0.46t^2 - 366t - 1.85 = 0
t = 8.43 s. To catch up.
B. d = 0.5a*t^2 = 2.27*8.43^2 = 161.3 m.
C. V = a*t = 4.53*8.43 = 38.2 m/s.
D. 3.62*8.43 = 30.5 m/s.
To find the answer to these questions, we need to use the equations of motion for uniformly accelerated motion. These equations relate displacement, velocity, acceleration, and time. Here are the steps to solve each part:
(A) To find the time it takes Kathy to overtake Stan, we need to find the time it takes for both of them to cover the same displacement. Since Kathy starts 1.01 seconds after Stan, we can calculate the time it takes for both cars to reach the same position.
Let t be the time taken by Kathy to overtake Stan.
For Kathy:
Displacement = (1/2) * acceleration * t^2 + initial velocity * t
For Stan:
Displacement = (1/2) * acceleration * (t + 1.01)^2 + initial velocity * (t + 1.01)
Set the two displacement equations equal to each other and solve for t:
(1/2) * 4.53 * t^2 + 0 * t = (1/2) * 3.62 * (t + 1.01)^2 + 0 * (t + 1.01)
4.53 * t^2 = 3.62 * (t + 1.01)^2
Simplify the equation:
4.53 * t^2 = 3.62 * (t^2 + 2.02t + 1.0201)
Expand and rearrange the equation:
4.53 * t^2 = 3.62 * t^2 + 7.3344t + 3.650562
0.91 * t^2 - 7.3344t - 3.650562 = 0
Now we can solve this quadratic equation to find the value of t. You can use the quadratic formula or factorization method to solve it.