Both start from rest, but experienced Stan leaves the starting line 1.00 s before Kathy. Assume Stan moves with a constant acceleration of 3.45 m/s2 and Kathy maintains an acceleration of 5.10 m/s2.
(a) Find the time it takes Kathy to overtake Stan.
(b) Find the distance she travels before she catches him.
(c) Find the speeds of both cars at the instant she overtakes him.
Kathy's speed =
Stan's speed=
To find the time it takes Kathy to overtake Stan, we need to determine when they meet. Since both start from rest, we can use the equation:
Displacement = Initial velocity * Time + 0.5 * Acceleration * Time^2
For experienced Stan, his initial velocity is 0 (since he starts from rest), and his acceleration is 3.45 m/s^2. Therefore, the equation for Stan becomes:
Displacement = 0 * Time + 0.5 * 3.45 * Time^2
For Kathy, her initial velocity is also 0, but her acceleration is 5.10 m/s^2. Therefore, the equation for Kathy becomes:
Displacement = 0 * (Time + 1) + 0.5 * 5.10 * (Time + 1)^2
Since they meet at the same displacement, we can set these two equations equal to each other and solve for the time when they meet:
0 * Time + 0.5 * 3.45 * Time^2 = 0 * (Time + 1) + 0.5 * 5.10 * (Time + 1)^2
Simplifying this equation gives:
1.725 * Time^2 = 2.55 * (Time^2 + 2Time + 1)
1.725 * Time^2 = 2.55 * Time^2 + 5.1 * Time + 2.55
0.825 * Time^2 - 5.1 * Time - 2.55 = 0
To solve this quadratic equation, we can use the quadratic formula:
Time = (-b ± √(b^2 - 4ac)) / 2a
For this equation, a = 0.825, b = -5.1, and c = -2.55. Plugging these values into the quadratic formula gives us the time it takes Kathy to overtake Stan.
(a) To find the time it takes Kathy to overtake Stan, we solve for Time using the quadratic formula.