Question

Cars a and b travel in a straight line. the distance of a from the starting point is given as a function of time by xa1t2 = at bt 2 , with a = 2.60 m>s and b = 1.20 m>s 2 . the distance of b from the starting point is xb1t2 = gt 2 - dt 3 , with g = 2.80 m>s 2 and d = 0.20 m>s 3 . (a) which car is ahead just after the two cars leave the starting point?

Answer 1

Just after leaving the starting point, Car A is ahead because it has an initial velocity while Car B starts from rest with acceleration, and the higher-order terms are negligible at t just after 0.

Step-by-step explanation:

To determine which car is ahead just after leaving the starting point, we must consider the position functions of cars A and B at time t just after t = 0. The position of Car A is given by xa(t) = at + bt2, with a = 2.60 m/s and b = 1.20 m/s2. The position of Car B is xb(t) = gt2 - dt3, with g = 2.80 m/s2 and d = 0.20 m/s3. Just after t = 0, we can ignore the t-squared and t-cubed terms because they are negligible, and compare the linear terms. Car A does not have a t-squared term immediately after leaving the start, while Car B does not have a linear term.

Thus, Car A is ahead just after the two cars leave the starting point because its initial velocity (a) contributes to its position from the very start, while Car B has no initial velocity and starts with acceleration (which does not have an immediate effect on position).