Question

Consider two toy cars. Car A starts from rest and speeds up with constant acceleration for a time tA until it reaches a speed of vA 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 aB.
a) Make a freehand plot of the speed of each car as a function of time. Label your plot with all relevant information, including any given slope information.
c) Determine how much time, ∆t, elapses between the time car B starts moving and the time car B passes car A. To do this, you will need to solve a quadratic equation. Explain your choice of sign for the radical.
2
b) Make a freehand plot of the position of each car as a func- tion of time. Label your plot with all relevant information, including any given slope information.
d) Determine the ratio vB/vA, where vB is the speed of car B at the moment it passes car A. Simplify your answer as much as possible. What is the limit of vB/vA as aB → 0? Will car B always eventually overtake car A no matter how small we make aB (as long as aB is nonzero)?

Answer 1

a)

For A For B Vlt) - - - - - - - - - T +A

c)

let the car B catch car A after time $\Delta$t

Consider the motion of car A :

V_Ao = initial velocity of car A = 0 m/s

t_A = time during which the car accelerate

$\Delta$t = time during which the car A moves at constant speed

Position of car A is given as

X_A = V_Ao t_A + (0.5) a_A t_A² + V_A$\Delta$t

X_A = (0) t_A + (0.5) a_A t_A² + V_A$\Delta$t

X_A = V_A$\Delta$t + (0.5) a_A t_A²                                                      eq-1

Consider the motion of car B :

V_Bo = initial velocity of car B = 0 m/s

$\Delta$t = time during which the car B accelerate

a_B = acceleration of car B

Position of car B is given as

X_B = V_Bo$\Delta$t + (0.5) a_B$\Delta$t²

X_B = (0) $\Delta$t + (0.5) a_B$\Delta$t²

X_B = (0.5) a_B$\Delta$t²                                                      eq-2

Using eq-1 and eq-2, for the cars to catch each other

X_A = X_B

V_A$\Delta$t + (0.5) a_A t_A² = (0.5) a_B$\Delta$t²

(0.5) a_B$\Delta$t² - V_A$\Delta$t - + (0.5) a_A t_A² = 0

$\Delta$t = (- (- V_A) $\pm$ sqrt((- V_A)² - 4 ((0.5) a_B ) ((0.5) a_A t_A²)) )/a_B

$\Delta$t = ( V_A$\pm$ sqrt( V_A² - a_B a_A t_A² )/a_B

the sign will be positive since time can not be negative