8Dvelocitygraphs


 * Velocity-Time Graphs**

**Acceleration** at time, t, is the gradient of the velocity-time graph. **Displacement** is the definite integral (or the signed area) between the graph and the time axis. **Distance travelled** is the sum of the unsigned areas between the graph and the time axis.


 * Example 1 **

Consider the following velocity-time graph

a) Find the acceleration in the first 10 seconds acceleration = gradient (0 to 10)

math \\ . \qquad a = \frac{10}{10} \\. \\ . \qquad a =1 \; m/s^2 math

b) Find the displacement after 10 seconds displacement = area (0 to 10)

math \\ . \qquad s=\frac{1}{2} \times 10 \times 10 \\. \\ . \qquad s=50 m math

c) Find the displacement after 25 seconds displacement = total area (0 to 25)

math \\ . \qquad s=50+100+25 \\. \\ . \qquad s=175 m math

d) Find the distance travelled after 40 seconds distance travelled = total unsigned area (0 to 40)

math \\ . \qquad \text{distance }=175+75 \\. \\ . \qquad \text{distance }=250 m math

e) Find the displacement after 40 seconds displacement = total signed area (0 to 40)

math \\ . \qquad s= 175-75 \\. \\ . \qquad s=100 m math {object finishes at rest, 100 metres to the right of where it started}


 * Example 2 **

A body moves with a velocity (m/s) specified by: math . \qquad v(t)= \frac{1}{4}t^2 - \frac{1}{2}t - 2 \quad \text{ for } \; t \in \big[ 0, 7 \big] math

a) Sketch the velocity-time graph. b) Find the displacement after 7 seconds. c) Find the distance travelled in the first 7 seconds d) Find the average velocity over the first 7 seconds e) Find the average speed over the first 7 seconds

__**Solution**__


 * a) Sketch Velocity-Time graph **


 * b) Find the displacement after 7 seconds **

displacement = signed area from 0 to 7 math \\ . \qquad s = \displaystyle{\int\limits_0^7} \, \frac{1}{4}t^2 - \frac{1}{2}t - 2 \, dt \\. \\ . \qquad s = \Big[ \frac{1}{12}t^3 - \frac{1}{4}t^2 - 2t \Big]_0^7 \\. \\ . \qquad s = \frac{7}{3} \; \text{ metres} math


 * c) Find the distance travelled after 7 seconds **

distance travelled = total unsigned area from 0 to 7 math \\ . \qquad \text{dist } = - \displaystyle{\int\limits_0^4} \, \frac{1}{4}t^2 - \frac{1}{2}t - 2 \, dt + \displaystyle{\int\limits_4^7} \, \frac{1}{4}t^2 - \frac{1}{2}t - 2 \, dt \\. \\ . \qquad \text{dist } = - \Big[ \frac{1}{12}t^3 - \frac{1}{4}t^2 - 2t \Big]_0^4 + \Big[ \frac{1}{12}t^3 - \frac{1}{4}t^2 - 2t \Big]_4^7 \\. \\ . \qquad \text{dist } = \dfrac{47}{3} \; \text{ metres} math


 * d) Find the average velocity over the first 7 seconds **

average velocity = __displacement__ divided by time math \\ . \qquad \text{ave vel } = \dfrac{ \frac{7}{3} }{7} \\. \\ . \qquad \text{ave vel } = \dfrac{1}{3} \; m/s math


 * e) FInd the average speed over the first 7 seconds **

average speed = __distance travelled__ divided by time math \\ . \qquad \text{ave speed } = \dfrac{ \frac{47}{3} }{7} \\. \\ . \qquad \text{ave speed } = \dfrac{47}{21} \ m/s math

.