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The acceleration time diagram for a linear motion is shown. Construct velocity time diagram and displacement time diagram for the motion assuming that the motion starts - Engineering Mechanics

Answer in Brief

The acceleration time diagram for a linear motion is shown. Construct velocity time diagram and displacement time diagram for the motion assuming that the motion starts with a initial velocity of 5 m/s from the starting point.

Given : Acceleration time graph
To draw : Velocity time graph
               Displacement time graph

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Solution

FOR VELOCITY TIME GRAPH :
We know that the area under a-t graph gives the velocity.
AB on a-t graph represents linearly varying deceleration
`v_0 = 5  m ⁄ s `

`v_1 = v_0 + A(△ OAB )`
`= 5 + 1/2 xx 6 xx 1`
= 8 m/s
BC on a-t graph represents linearly varying acceleration

`v_2 = v_1 + A(△BCD)`
`=8 + 1/2 xx (12-6) xx 2`
=14 m/s
The velocity time graph is drawn below :

FOR DISPLACEMENT TIME GRAPH :
Area under v-t graph gives the displacement

Area under EF = A(EFH) + A(▭EHIO)
`=2/3 xx 6 xx (8-5) + 6 xx 5`
= 42 m

Area under FG = A(GFH) +A(▭FIJK)
`=1/3 xx (12 – 6) xx (14-8) + (12 – 6) xx 8`
=12 + 48
= 60 m
The displacement time graph is shown below :

Concept: Velocity and Acceleration in Terms of Rectangular Co-ordinate System
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