Derive the equation of Motion using graphical method.
Answers
Answer:
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Explanation:
The graph of velocity against time is drawn for an object that starts moving with an initial velocity u and moves with certain acceleration a and travels for time t and has a final velocity v when observed. Now we all know that the slope of the curve of a velocity-time graph gives the acceleration of the particle and the area under the curve gives us the displacement of the particle.
The curve or the plot of the motion is given by the line AB; and since the co-ordinates of the points A and B are known, its slope can be given as
m=(y2−y1)(x2−x1)
Now we know that this slope gives us the value of the acceleration of the particle, hence we can say
a=(v−u)(t−0)⇒at=v−u⇒v=u+at
This is the first equation of motion. Now the second equation of motion involves the displacement of the particle and we have discussed above that the area under the curve gives the displacement of the particle. Using this approach, we can say that the area of trapezium AOBC gives the displacement. We know that the area of a trapezium is given as half the product of the sum of the parallel sides and the distance between them.
For the given graph, we can say that
displacement=12×(OA+BC)×(OC)
Substituting the values of the mentioned sides, we get
s=12×(u+v)×(t)−−equation(a)
Substituting the value of the final velocity from the first equation and simplifying, we get
s=12×(u+u+at)×(t)⇒s=12×(2ut+at2)⇒s=ut+12at2
This is the second equation of motion.
From the first equation of motion, we can say that t=(v−u)a
Substituting this value of time in equation (a) obtained above, we get
s=12×(u+v)×(v−u)a⇒2as=(v+u)(v−u)⇒2as=v2−u2[∵(a+b)(a−b)=a2−b2]⇒v2=u2+2as
This is the third equation of motion.
Thus we have successfully derived all the equations using the graphical method.
Note
We have three methods of deriving the equations of motion, namely the algebraic method, the graphical method and the calculus method. The algebraic method can be related very closely to the graphical method that we used in our solution above. The calculus method is used when the object changes its state over infinitesimal intervals of time and hence depicting the motion by a graph is not feasible.