Mathematics

Problem

Let

s(t)

denote the vertical displacement of a body in free fall (downward positive) and

v(t)

its instantaneous velocity. Answer the following questions without assuming any specific numerical values. a. (i) Starting from the definition of instantaneous velocity, show that the slope of the position–time graph at any instant equals the ordinate of the velocity–time graph at the same instant. b. (ii) Prove that the area under the velocity–time graph between times

t_1

and

t_2

equals the change in displacement

s(t_2)-s(t_1)

. c. (iii) For motion with constant acceleration

g

and initial velocity

u=0

, use parts (i) and (ii) to derive the algebraic relation

s=\tfrac12 g t^2

Intuitive explanation

The key insight is that velocity and displacement are connected through calculus: $v(t)$ is the derivative of $s(t)$ , and acceleration is the derivative of $v(t)$ . In free fall, gravity provides a constant downward acceleration, so you can build $v(t)$ and $s(t)$ by integrating this constant and applying initial conditions. Understanding these relationships lets you answer questions about motion without plugging in numbers—you work symbolically with the structure of the problem.

Intuitive explanation

Browse the problem library