### Q.23:- **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})

**Answer:-**

(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.