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.

---

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

---

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.

---

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. 

---

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