Rain appears to fall vertically on a man walking at 3 kmh -1 but if he doubles his speed to the rain appears to fall at 45 o . The real velocity of rain is

3√2 kmh -1 , 45 o with vertical in clockwise direction

2√3 kmh -1 , 45 o with vertical in clockwise direction

3√2 kmh -1 , 45 o with vertical in anticlockwise direction

Important Relative Motion Formulas for JEE

Relative motion is a combined property of the object under study as well as the observer. There is no such thing as absolute motion or absolute rest. Motion is always defined with respect to an observer or reference frame. Having said that, here on this page, students will find a list of relative motion formulas which will further boost their revision before the entrance exams.

Relative Motion in One Dimension

1. Relative Position

\(\begin{array}{l}\vec{X}_{AB}=\vec{X}_{A}-\vec{X}_{B}\end{array} \)

Here,

\(\begin{array}{l}\vec{X}_{A}\ \text{and}\ \vec{X}_{B}\ \text{are the position vector of A and B with respect to the origin.}\end{array} \)

2. Relative Velocity

\(\begin{array}{l}\vec{V}_{AB}=\vec{V}_{A}-\vec{V}_{B}\end{array} \)

Here,

\(\begin{array}{l}\vec{V}_{A}\ \text{and}\ \vec{V}_{B}\ \text{are the velocity w.r.t to ground}.\end{array} \)

\(\begin{array}{l}\vec{V}_{AB}\ \text{is the velocity of A w.r.t B.}\end{array} \)

All Velocities are relative and will not have any significance unless the observer is specified. However, when we say “velocity of A” it means the velocity of A w.r.t to the ground, which is assumed to be at rest.

3. Relative Acceleration

It is the rate at which relative velocity is changing

\(\begin{array}{l}\vec{a}_{AB} = \frac{d}{dt} \vec{V_{AB}} = \frac{d}{dt} \vec{V_{A}} = \frac{d}{dt} \vec{V_{B}}\end{array} \)

4. Equation of motion (if relative acceleration is constant)

v = u + at

S = ut + (1/2)at²

v² = u²+ 2as

5. Velocity of Approach/Separation

It is the component of the relative velocity of one particle w.r.t another along the line joining them. If the separation is decreasing, we say it is the velocity of approach, and if separation is increasing, then we say it is the velocity of separation.

Relative Motion in two Dimension

Relative Motion in two dimension

\(\begin{array}{l}\vec{r}_{AB}= \vec{r}_{A}-\vec{r}_{B}\end{array} \)

Here,

\(\begin{array}{l}\vec{r}_{A},\vec{r}_{B}\ \text{are position vector of A and B w.r.t origin}.\end{array} \)

\(\begin{array}{l}\vec{V}_{AB}=\vec{V}_{A}-\vec{V}_{B}\end{array} \)

Here,

\(\begin{array}{l}\vec{V}_{A}\ \text{and}\ \vec{V}_{B}\ \text{are the velocity w.r.t to ground}.\end{array} \)

\(\begin{array}{l}\vec{a}_{AB} = \frac{d}{dt} \vec{V_{AB}} = \frac{d}{dt} \vec{V_{A}} – \frac{d}{dt} \vec{V_{B}}\end{array} \)

\(\begin{array}{l}=\vec{a}_{A}-\vec{a}_{B}\end{array} \)

Here,

\(\begin{array}{l}\vec{a}_{A}, \vec{a}_{B},\ \text{are the acceleration w.r.t to ground}\end{array} \)

Relative Motion in River Flow

Consider a man swimming in a river with velocity V_MR relative to the river at an angle of θ with the river flow. The velocity of the river is V_R. Let us consider 2 observers (observer 1 and observer 2). Observer 1 is on the ground, and observer 2 is in a raft floating along with the river. Therefore the motion w.r.t to observer 2 is the same as the motion w.r.t to the river. The man will flow with angle θ with the river flow for observer 2. For observer 1, the velocity of the river will be V_M=V_MR+ V_R.Therefore the swimmer will appear to move at an angle θ’ with the river flow.

Relative Motion in river flow

1. Drift

It is defined as the displacement of the man in the direction of the river flow. In the figure above x is the drift.

2. Crossing the river in the shortest time

Time to cross the river will be minimum if man swims perpendicular to the river flow

T_min = d/V_MR

3. Crossing the river in the shortest path, Minimum drift

The minimum possible drift is zero. In this case, the man swims in the direction perpendicular to the river flow as seen from the ground. This path is known as the shortest path. For minimum drift, the man must swim at some angle Φ with the perpendicular in a backward direction, which is given by sin Φ = V_R/V_MR

Time to cross the river along the shortest path

t = d/V_MRcos Φ

\(\begin{array}{l}t=\frac{d}{\sqrt{V^{2}_{MR}-V^{2}_{R}}}\end{array} \)

Wind Airplane Problem

The velocity of the aeroplane with respect to the wind is

\(\begin{array}{l}\vec{V}_{AW}=\vec{V}_{A}-\vec{V_{W}}\end{array} \)

Here

\(\begin{array}{l}\vec{V}_{A}\ \text{is the aeroplane w.r.t the ground,}\end{array} \)

\(\begin{array}{l}\vec{V}_{W}\ \text{is the velocity of the wind.}\end{array} \)

Rain Problem

The rain is falling vertically with a velocity V_Rand the observer is moving horizontally with velocity V_M,the velocity of rain relative to an observer will be

\(\begin{array}{l}\vec{V}_{RM}=\vec{V_{R}}-\vec{V}_{M}\end{array} \)

And direction θ = tan^-1(V_m/V_R)

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