Question

For any arbitrary motion in space, which of the following relations are true:
A. ${v_{average}} = \left( {\dfrac{1}{2}} \right)\left( {v\left( {{t_1}} \right) + v\left( {{t_2}} \right)} \right)$
B. ${v_{average}} = \dfrac{{\left[ {r\left( {{t_2}} \right) - r\left( {{t_1}} \right)} \right]}}{{{t_2} - {t_1}}}$
C. $v\left( t \right) = v\left( 0 \right) + at$
D. $r\left( t \right) = r\left( 0 \right) + v\left( 0 \right)t + \dfrac{1}{2}a{t^2}$
E. ${a_{average}} = \dfrac{{\left[ {v\left( {{t_2}} \right) - v\left( {{t_1}} \right)} \right]}}{{{t_2} - {t_1}}}$
(The average stands for average of the quantity over the time interval ${t_1}$ to ${t_2}$ )

Answer 1

Hint:Here we have to examine each of the options one by one to get the answer. An object's average velocity is the total displacement separated by the total time taken. In other words, it is the pace at which an object from one location to another changes its position. A vector quantity is average velocity. The pace at which velocity varies is the average acceleration. The change of velocity divided by an elapsed duration is the average acceleration.

Complete answer:
Here the velocities and the acceleration should be valid for both uniform and non-uniform motion. A body is assumed to have variable velocity if it covers the unequal distance in a given direction at equal time intervals, but the interval might be short or it covers the same distance at equal time intervals, but the direction tends to change.
If we observe the option A we can see that the equation does not satisfy non uniform acceleration. Hence, for any arbitrary motion in space it is false.
Now if we see option B we can see that the equation satisfies both uniform and non-uniform average velocity. Hence, it is true for any arbitrary motion in space.
For option C the initial velocity is zero. So it is not true to uniform motion. Hence, it is also false for any arbitrary motion in space.The option D equation is also invalid for arbitrary motion in space. Hence, it is false.
The option E equation satisfies both uniform and non-uniform acceleration. Hence, it is true for any arbitrary motion in space.

Thus, only options B and E are true.

Note:Here we have to know what uniform and non-uniform motion is only then we will be able to answer. When a body travels in a straight line and crosses equal lengths at equal periods of time, the motion of a particle or a body is considered to be uniform. If the body covers unequal distances at equal intervals of time or covers equal distances at unequal intervals of time, the motion of a particle or a body is considered to be non-uniform.