Consider two toy cars. Car A starts from rest and speeds up with constant acceleration for a time delta t until it reached a speed of v and then continues to travel at this speed. At the moment car A reaches its maximum speed, car B, starting at rest from the same point that car A started from, speeds up with constant acceleration.
Determine the ratio Vb/Va where is the speed of car B at the moment it passes car A. Simplify your answer as much as possible. What it the limit Vb/Va as acceleration B approaches 0?
To determine the ratio Vb/Va, we need to first understand the motion of both cars.
For Car A:
Since Car A starts from rest and speeds up with constant acceleration, we can use the kinematic equation:
v = u + a*t
where:
v = final velocity (which is V for Car A)
u = initial velocity (which is 0 for Car A since it starts from rest)
a = acceleration
t = time taken to reach final velocity (delta t for Car A)
Since Car A reaches a speed of V, we have:
V = 0 + a*(delta t)
This can be rearranged to solve for a:
a = V / (delta t)
Now, let's move on to Car B:
Car B also starts from rest and speeds up with constant acceleration. Similar to Car A, we can use the same kinematic equation:
v = u + a*t
where:
v = final velocity (which is Vb for Car B)
u = initial velocity (which is 0 for Car B since it starts from rest)
a = acceleration
t = time taken to reach final velocity
Since Car B passes Car A at the moment Car A reaches its maximum speed, we can assume that both cars reached their respective speeds at the same time. Therefore, the time taken for Car B to reach its speed (Vb) is also delta t.
Hence, we have:
Vb = 0 + a*(delta t)
This can be rearranged to solve for a:
a = Vb / (delta t)
Now, we can find the ratio Vb/Va by dividing the expression for Car B by the expression for Car A:
(Vb / (delta t)) / (V / (delta t))
The delta t terms cancel out, resulting in:
Vb / Va
This is the ratio Vb/Va at any given time.
Now, let's consider the limit as acceleration B approaches 0. In this case, as acceleration B gets smaller and smaller, the ratio Vb/Va approaches infinity since Vb will keep getting larger while Va remains constant at V. Therefore, the limit of Vb/Va as acceleration B approaches 0 is infinity.