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

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

(a) $\mathbf{v}_{\text {aventes }}=\left(\frac{1}{2}\right)\left(\mathbf{v}\left(t_{1}\right)+\mathbf{v}\left(t_{2}\right)\right)$

(b) $\mathbf{v}_{\text {avenee }}=\frac{\left[\mathbf{r}\left(t_{2}\right)-\mathbf{r}\left(t_{1}\right)\right]}{\left(t_{2}-t_{1}\right)}$

(c) $\mathbf{v}(t)=\mathbf{v}(0)+\mathbf{a} t$

(d) $\mathbf{r}(t)=\mathbf{r}(0)+\mathbf{v}(0) t+\left(\frac{1}{2}\right) \mathbf{a} t^{2}$

(e) $\mathbf{a}_{\text {averace }}=\frac{\left[\mathbf{v}\left(t_{2}\right)-\mathbf{v}\left(t_{1}\right)\right]}{\left(t_{2}-t_{1}\right)}$

(The 'average' stands for average of the quantity over the time interval $t_{1}$ to $t_{2}$ )

solution:

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

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

(c)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.

(d)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.

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