Graph
- The velocity-displacement graph of a particle is given. Plot acceleration-displacement graph of it.
To plot acceleration-displacement graph we need an equation of acceleration and displacement from the given graph. Using equation of straight line (y = mx + c) we get, v = (-
)x + v0. Slope of the graph is m = – (angle is in clockwise direction).
Acceleration a = v
= (- 
+ v0).(-
) [as = –
]
So, a = 
.x – 
[comparing the equation y = mx + c, m = + ve and c = – ve].
CONCEPTS ON GRAPH:
- A ball is released from a height under gravity. It collides with the ground and during each collision it looses its speed. Taking downward motion positive plot (i) distance – time, (ii) displacement – time, (iii) speed – time, (iv) velocity – time and (v) acceleration – time graph.
Let us consider the ball is dropped from a height s1 and it takes t1 time to reach the ground. Before collision on ground its speed is v1. After collision it moves with speed v2 and travels for time t2 to reach maximum height s2. Again when it collides with ground, after collision it moves with speed v3 and travels for time t3 to reach maximum height s3.
(i) Distance – time graph: Downward motion of the ball is accelerated motion [slope positive] and upward motion is retarded motion [slope negative].
(ii) Displacement – time graph: When the ball reaches the maximum height after 1st collision the displacement is s1 – s2. Similarly when the ball reaches the maximum height after 2ndcollision the displacement is s1 – s3. Finally when the ball becomes rest, the displacement is s1.
(iii) Speed – time graph: As speed is a scalar quantity it is represented in 1st quadrant and downward motion of the ball is accelerated motion [slope positive and constant] and upward motion is retarded motion [slope negative and constant].
(iv) Velocity – time graph: As velocity is a vector quantity, during downward motion of the ball velocity is positive and upward motion it is negative.
(v) Acceleration – time graph: As acceleration is a vector quantity, during downward motion of the ball acceleration is positive (a = +g)and upward motion it is negative (a = -g).
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