Question

A particle moves in a straight line with uniform acceleration. Its velocity at time t=0 and at time t=t is V. The average velocity of the particle in this time interval is:

Answer 1

Hint: In this question, we will use the required equation of motion, which gives us the relation between velocity, acceleration and distance of an object. Further, by substituting the values in the basic velocity equation, will give us the required result. Also, we will discuss the basics of the equations of motion, for our better understanding.

Formula used:
 ${v_{avg}} = \dfrac{s}{t}$
${v^2} - {u^2} = 2as$

Complete step-by-step answer:
As we know, acceleration is the rate of change of velocity of an object with time. We can write as:
$a = \left( {\dfrac{{{v_2} - {v_1}}}{t}} \right)$
$ \Rightarrow t = \left( {\dfrac{{{v_2} - {v_1}}}{a}} \right)$
Now, we will use the equation of motion, which is given as:
$v_2^2 - v_1^2 = 2as$
Also, we know that, velocity is given by the ratio of speed and time, we can write it as:
${v_{avg}} = \dfrac{s}{t}$
Now, substituting the values of distance s and time t, in the above equation, we get:
${v_{avg}} = \dfrac{{v_2^2 - v_1^2}}{{2a\left( {\dfrac{{{v_2} - {v_1}}}{a}} \right)}}$
$\therefore {v_{avg}} = \dfrac{{{v_2} + {v_1}}}{2}$

Additional Information: As we know that the equations of motion are equations which describe the behavior of a physical system in terms of its motion as a function of time. Further we can say that these equations of motion describe the behavior of a physical system as a set of mathematical functions in terms of dynamic variables. Here, dynamic variables are said to be normally spatial coordinates and time is used, but others are also possible, like momentum components and time.
Now, if we go in history, these equations of motion were discovered by Galileo Galilee but he could not manage to prove it practically that his equations were right or not. Later, Sir Isaac Newton proved these three equations of motion practically and also graphically. So, that is the reason now they are often called Newton’s three equations of motion. These equations tell us about the acceleration, displacement, time, final velocity of an object, initial velocity of an object.

Note: Here we should remember that the three different equations of motion are used in finding different physical properties of a particle under motion. We should also observe that these equations are only applicable to the classical system not in the quantum system.