#### Question

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

a) `V_"average"` = (1/2) (v (t_{1}) + v (t_{2}))

b) `V_"average"` = [r(t_{2}) - r(t_{1}) ] /(t_{2} – t_{1})

c) v (t) = v (0) + a t

d) r (t) = r (0) + v (0) t + (1/2) a t^{2}

e) `a_"average"` = =[ v (t_{2}) - v (t_{1} )] /( t_{2} – t_{1})

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

#### Solution 1

(b) and (e) are true; others are false because relations (a), (c) and (d) hold only for uniform acceleration.

#### Solution 2

**(b)** and **(e)**

**(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.