# How to Interpret the Displacement-Time Graphs

## Question 1

(a)     Find the initial displacement.

$x=0$

(b)     Find the displacement at $t=4$.

$x=3$

(c)     Find the velocity during $0 \le t \le 3$.

$\displaystyle \text{velocity} = \frac{\text{rise}}{\text{run}} = \frac{3-0}{3-0} = 1 \text{ ms}^{-1}$

(d)     Find the velocity during $3 \le t \le 7$.

$\displaystyle \text{velocity} = \frac{\text{rise}}{\text{run}} = \frac{3-3}{7-3} = \frac{0}{4} = 0 \text{ ms}^{-1}$

(e)     Find the velocity during $7 \le t \le 10$.

$\displaystyle \text{velocity} = \frac{\text{rise}}{\text{run}} = \frac{0-3}{10-7} = -1 \text{ ms}^{-1}$

(f)     Find the distance travelled for $0 \le t \le 10$.

$3+0+3=6 \text{ m}$

## Question 2

(a)     Find the initial displacement.

$x=3$

(b)     Find the displacement at $t=4$.

$x=-1$

(c)     Find the velocity during $0 \le t \le 5$.

$\displaystyle \text{velocity} = \frac{\text{rise}}{\text{run}} = \frac{-2-3}{5-0} = -1 \text{ ms}^{-1}$

(d)     Find the time(s) when the velocity becomes zero.

$5 \le t \le 8$

(e)     Find the time(s) when the velocity becomes positive.

$8 \le t \le 10$

(f)     Find the time(s) when the particle returns to the origin.

$t=3 \text{ and } 10 \text{ seconds}$

(g)     Find the time(s) when the particle moves backward.

$8 \le t \le 10$

(h)     Find the distance travelled for $0 \le t \le 10$.

$5+0+2=7 \text{ m}$

## Question 3

(a)     Find the time(s) when the particle changes its direction.

$t=2 \text{ and } 6 \text{ seconds}$

(b)     Find the distance travelled for the first $10$ seconds.

$3+6+3=12 \text{ m}$

(c)     Find the velocity during $0 \le t \le 3$.

$\displaystyle \text{velocity} = \frac{\text{rise}}{\text{run}} = \frac{3-0}{3-0} = 1 \text{ ms}^{-1}$

(d)     Find the velocity during $3 \le t \le 6$.

$\displaystyle \text{velocity} = \frac{\text{rise}}{\text{run}} = \frac{-3-3}{6-3} = -2 \text{ ms}^{-1}$

(e)     Find the velocity during $6 \le t \le 10$.

$\displaystyle \text{velocity} = \frac{\text{rise}}{\text{run}} = \frac{3}{4} \text{ ms}^{-1}$

(f)     Find the maximum speed of the particle.

Speed is the positive value of velocity, thus $2 \text{ ms}^{-1}$.

## Question 4

(a)     Find the time(s) when the particle changes its direction.

$t=2 \text{ and } 6 \text{ seconds}$

(b)     Find the distance travelled for the first $10$ seconds.

$4+8+4=16 \text{ m}$

(c)     Find the time(s) when the particle’s velocity becomes maximum.

velocity is the gradient of $x$ maximum gradient occurs at $t=4$.

(d)     Find the time(s) when the particle moves forward.

$2 \lt y \lt 6$

## Question 5

(a)     Find the speed when the particle reaches the origin.

$1 \text{ms}^{-1}$

(b)     Find the distance travelled for the first $10$ seconds.

$10 \text{ m}$

## Question 6

(a)     Find the time(s) when the particle changes its direction.

$t=2 \text{ and } 8 \text{ seconds}$

(b)     Find the time(s) when the particle reaches its maximum velocity.

$t=0 \text{ and } 10 \text{ seconds}$ 