Name: Motion on an Inclined Plane
Uploaded: 2019-07-19
Duration: 6 min 34 s
Description: We all are curious about gravity. Through this video, let us learn to break the force of gravity into components in case of motion on an inclined plane.

Question

Show that the area enclosed by time-velocity graph and time-axis of an object moving with a constant acceleration in a straight line for a certain interval of time is equal to the distance travelled by the object in that time-interval.

Accepted Answer

If the velocity of an object is continuously changing at a definite rate, then the time-velocity graph will be a straight line inclined to the time-axis.

Suppose, at times $t_{1}$ and $t_{2}$ , the velocities of the object are $v_{1}$ and $v_{2}$ respectively. Then, below this line, the area between $t_{1}$ and $t_{2}$ is,

Area of triangle $A B C +$ Area of rectangle $A C B^{'} A^{'}$

$= \frac{1}{2} A C \times C B + A C \times A^{'} A$

$= \frac{1}{2} (t_{2} - t_{1}) \times (v_{2} - v_{1}) + (t_{2} - t_{1}) \times v_{1}$ or

$= [\frac{v_{1} + v_{2}}{2}] \times (t_{2} - t_{1}) = v_{a v} \times ∆ t$

According to definition, if during the time- interval $∆ t$ , the displacement of the object be $∆ s$ , then,
displacement $=$ average velocity x time-interval or
$∆ s = v_{a v} \times ∆ t$
Hence, by measuring this area, we can determine the displacement of the object. If the velocity of an object is changing in an irregular manner (at different rates), then its time velocity graph will be a curve. We can still determine the displacement of the object by calculating the area enclosed between the graph and the time axis.
It is clear from the above examples that the displacement of an object in a time-interval is equal to the area enclosed between time-velocity graph and time-axis for that time interval, whatever be the shape of the graph.

Show that the area enclosed by time-velocity graph and time-axis of an object moving with a constant acceleration in a straight line for a certain interval of time is equal to the distance travelled by the object in that time-interval.

Important Questions on Motion in One and Two Dimensions

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