Car A and B are facing each other at rest, 100 m apart. Car A accelerates at 5 m/s^2. Car B accelerates at 3 m/s^2. When and where do they meet?
To determine when and where Car A and Car B meet, we can use the equations of motion to analyze their positions and times at any given moment.
First, let's find the time it takes for both cars to meet. We'll assume that both cars start accelerating at the same time (t = 0) and meet at time t = t_meet.
Using the equation, s = ut + (1/2)at^2, where s is the displacement, u is the initial velocity, a is the acceleration, and t is time, we can determine the positions of the cars at any given time.
For Car A:
Initial position, s_A = 0 m
Initial velocity, u_A = 0 m/s
Acceleration, a_A = 5 m/s^2
For Car B:
Initial position, s_B = 100 m
Initial velocity, u_B = 0 m/s
Acceleration, a_B = 3 m/s^2
Now, we can calculate the position of each car at time t = t_meet:
s_A = (1/2)a_A(t_meet)^2
s_B = s_B + (1/2)a_B(t_meet)^2
Since we know that the sum of the distances traveled by both cars equals the initial separation distance between them:
s_A + s_B = 100 m
We substitute the position equations into the equation above:
(1/2)a_A(t_meet)^2 + s_B + (1/2)a_B(t_meet)^2 = 100 m
Simplifying the equation, we get:
(1/2)(5)(t_meet)^2 + 100 + (1/2)(3)(t_meet)^2 = 100
(5/2)t_meet^2 + (3/2)t_meet^2 = 0
Simplifying further, we have:
8t_meet^2 = 0
Solving for t_meet, we find that t_meet = 0.
This tells us that Car A and Car B meet at the instant they both start accelerating, meaning they will meet right at the starting point.
Therefore, Car A and Car B meet at time t = 0 and their meeting point is at the initial position of Car A, which is 0 m from the starting point.