A boy is on a train moving with a velocity of (50i + 30j) m/s relative to the roadside observer. The boy walks on the train with a velocity of (4i - 2j) m/s relative to the train. What is the velocity

The velocity of the boy relative to the train is $\vec{V}_{BT}$.

$$\vec{V}_{BT} = 4\hat{i} - 2\hat{j}$$

The velocity of the train relative to the roadside observer is $\vec{V}_{TR}$.

$$\vec{V}_{TR} = 50\hat{i} + 30\hat{j}$$

To find the velocity of the boy relative to the roadside observer ($\vec{V}_{BR}$), we add the velocity of the boy relative to the train and the velocity of the train relative to the roadside observer.

$$\vec{V}_{BR} = \vec{V}_{BT} + \vec{V}_{TR}$$

$$\vec{V}_{BR} = (4\hat{i} - 2\hat{j}) + (50\hat{i} + 30\hat{j})$$

Combine the $\hat{i}$ components and the $\hat{j}$ components:

$$\vec{V}_{BR} = (4 + 50)\hat{i} + (-2 + 30)\hat{j}$$ $$\vec{V}_{BR} = 54\hat{i} + 28\hat{j}$$

The velocity of the boy relative to the roadside observer is $54\hat{i} + 28\hat{j}$.