Question

An object has velocity $\overrightarrow{v_1}$ relative to the ground. An observer moving with a constant velocity $\overrightarrow{v_0}$ relative to the ground measures the velocity of the object as $\overrightarrow{v_2}$ (relative to the observer). The magnitude of these velocities are related by (where $v_0,v_1$ and $v_2$ are their respective magnitudes)

• A. $v_0\le v_1+v_2$
• B. $v_1\le v_2+v_0$
• C. $v_2\le v_0+v_1$
• D. All the above are true

Answer 1

The correct option is D All the above are true

The observer moving with velocity $\overrightarrow{v_o}$​​ relative to ground observes the maximum velocity of the object moving with velocity $\overrightarrow{v_1}$​​ , when both the object moves in the opposite direction relative to each other i.e. $\overrightarrow{v_2}=\overrightarrow{v_1}-(-\overrightarrow{v_0})=\overrightarrow{v_1}+\overrightarrow{v_0}$
For any other direction of motion between object moving with velocity $\overrightarrow{v_0}$ and object moving with velocity $\overrightarrow{v_1}$​​, the magnitude of relative velocity observed by the object moving with velocity $v_1$​​ will be less than $v_0+v_1$​.

Thus the magnitude of velocity of object moving with velocity $\overrightarrow{v_0}$ ​​relative to the object moving with velocity $\overrightarrow{v_1}$​​ will always be less than or equal to the sum of their velocity with respect to ground, that is
$v_2\le v_0+v_1$
Hence, the correct answer is option (d)