Question

A body is at rest at $x = 0$. At $t = 0$, it starts moving in the positive $x$-direction with a constant acceleration. At the same instant another body passes through $x = 0$ moving in the positive $x$-direction with a constant speed. The position of the first body is given by $x_1(t)$ after time $t$ and that of the second body by $x_2(t)$ after the same time interval. Which of the following graphs correctly describes $(x_1 - x_2)$ as a function of time $t$ ?

• A.

• B.

Answer 1

Answer: (B)

B

Answer 2

Let $x_1(t) = \frac{1}{2}at^2$ and $x_2(t) = vt$. The function to plot is $f(t) = x_1(t) - x_2(t) = \frac{1}{2}at^2 - vt$. This is a quadratic function $f(t) = At^2 + Bt$ with $A = a/2 > 0$ and $B = -v < 0$. Such a parabola opens upwards, passes through the origin at $t=0$, goes into negative values (since $B<0$ and $t>0$ initially, $Bt$ dominates $At^2$ for small $t$), reaches a minimum at $t = -B/(2A) = v/a$, and then increases. Graph (B) correctly depicts this behavior.