__Question 4.23__:

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})

__Solution__:

**Answer:**

**(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.

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