Question

For any arbitrary motion in space, which of the following relations are true :

1. $v_{average}$= $\bigg(\frac{1}{2}\bigg)$ ($v$ ($t_1$ ) + $v$ ($t_2$ ))
2. $v_{average}$ = $\frac{[r(t_2)-r(t_1)]}{(t_2-t_1)}$
3. $v(t)$ = $v(0)$ + $at$
4. $r(t)$ = $r(0)$ + $v(0)t+\bigg(\frac{1}{2}\bigg)at^2$
5. $a_{average}$ =$\frac{[v(t_2)-v(t_2)}{(t_2-t_1)}$
   (The ‘average’ stands for average of the quantity over the time interval $t_1$ to $t_2$ )

Answer 1

a) False
It is given that the motion of the particle is arbitrary. Therefore, the average velocity of the particle cannot be given by this equation.

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b) True
The arbitrary motion of the particle can be represented by this equation.

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c) False
The motion of the particle is arbitrary. The acceleration of the particle may also be non-uniform. Hence, this equation cannot represent the motion of the particle in space.

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d) False
The motion of the particle is arbitrary; acceleration of the particle may also be non-uniform. Hence, this equation cannot represent the motion of particle in space.

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e) True
The arbitrary motion of the particle can be represented by this equation.