Relative Velocity — Prelims Questions

Two cars, A and B, are moving in the same direction on a straight road. Car A is moving at $60\,\text{km/h}$ and Car B is moving at $45\,\text{km/h}$. What is the velocity of Car A relative to Car B?

A man is walking on a horizontal road at $3\,\text{m/s}$. Rain is falling vertically downwards with a speed of $4\,\text{m/s}$. At what angle with the vertical should the man hold his umbrella to protect himself from the rain?

A boat can travel at $10\,\text{m/s}$ in still water. The river flows at $6\,\text{m/s}$ due east. If the boat wants to cross the river directly to the north, what should be its resultant speed relative to the ground?

Two trains, each $100\,\text{m}$ long, are moving on parallel tracks. One train is moving at $72\,\text{km/h}$ and the other at $54\,\text{km/h}$. If they are moving in opposite directions, how long will it take for them to completely cross each other?

An airplane flies with a speed of $500\,\text{km/h}$ in still air. A wind is blowing from the west at $100\,\text{km/h}$. If the pilot wants to fly due north, what is the direction in which the pilot should head the airplane?

Two particles, P and Q, are moving with velocities $\vec{v}_P = (2\hat{i} + 3\hat{j}),\text{m/s}$ and $\vec{v}_Q = (4\hat{i} + \hat{j}),\text{m/s}$ respectively. What is the velocity of P relative to Q?

A particle moves with a constant velocity of $10\,\text{m/s}$ along the positive x-axis. Another particle starts from rest at the origin and accelerates at $2\,\text{m/s}^2$ along the positive x-axis. What is the relative acceleration of the first particle with respect to the second particle?