Name: Average Velocity
Uploaded: 2019-07-19
Duration: 3 min 57 s
Description: Unlike average speed, the average velocity of a moving object depends on the displacement, not the distance travelled. The video defines the formula for it.

Question

Derive the following equation of motion by the graphical method:

$v^{2} = u^{2} + 2 a s$

where the symbols have their usual meanings.

Accepted Answer

Consider the velocity-time graph of a body shown in the figure. The body has an initial velocity $v^{2} = u^{2} + 2 a s$ at point $A$ and then its velocity changes at a uniform rate from $A$ to $B$ in time $t$ . In other words, there is a uniform acceleration from $A$ to $B$ , and after time $t$ its final velocity becomes $v$ which is equal to $B C$ in the graph.To complete the figure, we draw the perpendicular $C B$ from point $C$ , and draw $A D$ parallel to $O C$ . $B E$ is perpendicular from point $B$ to $O E$ .

Suppose the body travels a distance $s$ in time $t$ . In figure, the distance travelled by the body is given by the area of the space between the velocity graph $A B$ and the time axis $O C$ , which is the area of the figure $O A B C$ , which is a trapezium. In other words:

$s =$ Area of trapezium $O A B C$

$s = \frac{(S u m o f p a r a l l e l s i d e s) \times H e i g h t}{2} s = \frac{(O A + C B) \times O C}{2}$

Now, $O A + C B = u + v$ and $O C = t$ . Putting these values in the above relation we get: $s = \frac{(u + v) \times t}{2} . . . . . . (1)$

We now want to eliminate $t$ from the above equation. This can be done by obtaining the value of $t$ from the first equation of motion.

Thus, $v = u + a t (F i r s t e q u a t i o n o f m o t i o n)$

$a t = v - u t = \frac{(v - u)}{a}$

Now, putting this value of $t$ in equation $(1)$ , we get,

$s = \frac{(u + v) (v - u)}{2 a}$

$2 a s = v^{2} - u^{2} [∴ (v + u) (v - u) = v^{2} - u^{2}] v^{2} = u^{2} + 2 a s$

This is the third equation of motion. It has been derived by the graphical method.

Derive the following equation of motion by the graphical method:

$v^{2} = u^{2} + 2 a s$

where the symbols have their usual meanings.

Important Questions on Motion

Learn and Prepare for any exam you want !

Derive the following equation of motion by the graphical method: v2=u2+2as where the symbols have their usual meanings.

Important Questions on Motion

Learn and Prepare for any exam you want !

Derive the following equation of motion by the graphical method:

$v^{2} = u^{2} + 2 a s$

where the symbols have their usual meanings.