Question

The plot below displays the velocity as a function of time for two physics students, Alan and Betty, for 10 seconds after a stopwatch is started at time t = 0. Alan and Betty are both initially standing at the same position, defined as

x = 0, and subsequently move along a straight line that we define as our x-axis. The plot for Betty’s velocity as a function of time vB(t) is itself a straight line while the plot for Alan is a quadratic (parabolic) function of time,

vA(t) = 2.0 - 0.8 t + 0.080 t2 (with t in seconds and v in m/sec).

vO) -2.0-0.8t + 0.08 0r (with r in seconds and v in m/sec ) 2.0T Alan 1.5 Betty 2 1.0 の0.5- 0.0 10 6 4 time (seconds) 0 2

A. Write the equation vB(t) for Betty’s velocity as a function of time over this 10 second interval.

B. What is Betty’s displacement, ∆x, during this 10 second interval?

C. What is Alan’s displacement during this interval?

D. What is Betty’s average acceleration, assuming she ends up traveling at 2.0 m/s as shown?

E. What is Alan’s average acceleration? F. At what time(s), if ever, does Alan have zero acceleration?

G. At what time(s), if ever, does Alan’s acceleration equal Betty’s acceleration?

H. At what time(s), if ever, does Alan’s velocity equal Betty’s velocity?

I. At what time(s), if ever, in this 10 second interval are Alan and Betty at the same position again after they leave x = 0? ________________________________

Answer 1

A. In this question we are given that the velocity of Alan is given by the relation 2 - 8t + 0.080t^2. We know that the Betty follows a linear path which should be given by the equation straight line. The equation of a straight line is given by v = mt + c. c is the point where line crosses the v axis and m is the slope of the line. Since c = 0, we can find the slope of the line by 2/10 = 0.2. so the equation of Betty will be v_B = 0.2t (1) B. We known that the velocity is given by the derivative of displacement v_B = dx/dt = 0.2t (2) by integrating this equation we can find the displacement x = 0.2t^2/2 = 0.2t^2 (3) for the time period of 10 seconds the displacement will be Delta x = 0.2 times 10^2 = 20m C. Similarly as above to find the displacement of the Alan we will integrate the equation of velocity of Alan.