Two cars are drag racing, starting from rest at t = 0, on a straight and level road. Team
Alpha's car produces acceleration
ax = alpha t^(-1/2) ;
while Team Beta's car produces acceleration
ax = Beta ;
for all t > 0, where alpha and beta are positive constants with suitable units.
a. Which car grabs the early lead? Justify answer.
b. If the race goes on long enough, the other car will overtake the early leader. Find
an expression for the distance x at which this will occur, in terms of the constants alpha
and Beta.
a. To determine which car grabs the early lead, we need to compare the positions of the two cars at a given time, t.
For Team Alpha's car, the acceleration is given by ax = alpha * t^(-1/2).
To find the position of Team Alpha's car, we need to integrate the acceleration with respect to time. Integrating ax = alpha * t^(-1/2) will give us the velocity of the car as a function of time:
vx = 2 * alpha * t^(1/2) + C1
(where C1 is the constant of integration)
Integrating the velocity with respect to time will give us the position of the car as a function of time:
x_alpha = (4/3) * alpha * t^(3/2) + C1 * t + C2
(where C2 is the constant of integration)
For Team Beta's car, the acceleration is constant and given by ax = Beta.
The velocity of Team Beta's car will then be given by integrating ax = Beta:
vx = Beta * t + C3
(where C3 is the constant of integration)
Integrating the velocity of Team Beta's car will give us the position as a function of time:
x_beta = (1/2) * Beta * t^2 + C3 * t + C4
(where C4 is the constant of integration)
To find the early leader, we need to compare the positions of the two cars at t = 0. When t = 0, both cars start from rest, so C1 = C2 = C3 = C4 = 0.
Comparing the initial positions, we have x_alpha = C2 = 0 and x_beta = C4 = 0.
Therefore, at t = 0, both cars have the same position, and no car has grabbed the early lead.
b. To determine when the other car will overtake the early leader, we need to find the time at which the positions of the two cars are equal.
Setting x_alpha = x_beta and solving for t:
(4/3) * alpha * t^(3/2) = (1/2) * Beta * t^2
Divide both sides by t^(3/2):
(4/3) * alpha = (1/2) * Beta * t
Solving for t:
t = (8/3) * (alpha / Beta)
Substituting this value of t back into the position equations for either car (x_alpha or x_beta) will give us the distance x at which the other car will overtake the early leader.