Physics > Motion in One Dimension > 7.0 Relative motion
Motion in One Dimension
1.0 Introduction
2.0 Kinematic variables
2.1 Distance and displacement
2.2 Average speed and velocity
2.3 Instantaneous speed and velocity
2.4 Average and instantaneous acceleration
3.0 Motion in one dimension
3.1 Motion in a straight line with uniform velocity
3.2 Motion in a straight line with uniform acceleration
3.3 Motion in a straight line with non-uniform acceleration
4.0 Derivation of the kinematics equation
5.0 Vertical motion under gravity
5.1 Basic terminologies for motion under gravity
5.2 Detailed concept of motion under gravity
5.3 Solved examples
6.0 Analysis of motion through graph
6.1 Displacement - time graph
6.2 Velocity - time graph
6.3 Area under the graph
6.4 Solved examples
7.0 Relative motion
7.1 Relative displacement
7.2 Relative velocity
7.3 Relative acceleration
7.4 Illustration of relative motion
7.5 Application of relative motion
8.0 Simultaneous motion of two bodies
9.0 River boat problem
9.1 Downstream
9.2 Upstream
9.3 Crosses the river in shortest interval of time
9.4 Reaches the point just opposite from where he started
9.5 River-man problem
9.6 Solved examples
10.0 Aircraft-wind problem
11.0 Rain problem
7.4 Illustration of relative motion
2.2 Average speed and velocity
2.3 Instantaneous speed and velocity
2.4 Average and instantaneous acceleration
3.2 Motion in a straight line with uniform acceleration
3.3 Motion in a straight line with non-uniform acceleration
5.2 Detailed concept of motion under gravity
5.3 Solved examples
6.2 Velocity - time graph
6.3 Area under the graph
6.4 Solved examples
7.2 Relative velocity
7.3 Relative acceleration
7.4 Illustration of relative motion
7.5 Application of relative motion
9.2 Upstream
9.3 Crosses the river in shortest interval of time
9.4 Reaches the point just opposite from where he started
9.5 River-man problem
9.6 Solved examples
A train is moving with velocity $v_1$ and a passenger in a train is moving with velocity $v_2$ wrt train. Study the motion of every body wrt observers $A$ and $B$.
Explanation: For vectors, it is essential to define our sign convention and unit vectors as below.
As we can see observer $A$ is standing on ground where as observer $B$ is moving along with the train.
We can write all velocities wrt ground as,
${\overrightarrow v _T} = + {v_1}\widehat i\ :$ Velocity of train
${\overrightarrow v _A} = 0\ :$ Velocity of observer $A$
${\overrightarrow v _B} = + {v_1}\widehat i\ :$ Velocity of observer $B$
${\overrightarrow v _{PT}} = + {v_2}\widehat i\ :$ Velocity of passenger wrt train
wrt observer $A$ (Ground frame) | wrt observer $B$ (Train frame) | |
| Velocity of train $\left( {{{\overrightarrow v }_T}} \right)$ | ${\overrightarrow v _{TA}} :$ Velocity of train wrt $A$, $${\overrightarrow v _{TA}} = {\overrightarrow v _T} - {\overrightarrow v _A}$$$${\overrightarrow v _{TA}} = {v_1}\widehat i - 0$$$${\overrightarrow v _{TA}} = {v_1}\widehat i = {\overrightarrow v _T}$$ As observer $A$ is at rest. So, observer $A$ will observe same velocity as observed from the ground frame of reference. | ${\overrightarrow v _{TB}}:$ Velocity of train wrt $B$$${\overrightarrow v _{TB}} = {\overrightarrow v _T} - {\overrightarrow v _B}$$$${\overrightarrow v _{TB}} = {v_1}\widehat i - {v_1}\widehat i$$$${\overrightarrow v _{TB}} = 0$$ |
| Velocity of passenger $\left( {{{\overrightarrow v }_P}} \right)$ | We know, $${\overrightarrow v _{PT}} = {v_2}\widehat i$$$${\overrightarrow v _A} = 0$$$${\overrightarrow v _T} = {v_1}\widehat i$$ Also, $${\overrightarrow v _{PT}} = {\overrightarrow v _P} - {\overrightarrow v _T}$$$${\overrightarrow v _P} = {\overrightarrow v _{PT}} + {\overrightarrow v _T}$$$${\overrightarrow v _P} = \left( {{v_2} + {v_1}} \right)\widehat i$$ ${\overrightarrow v _P}$ is the velocity of the passenger wrt ground frame $${\overrightarrow v _{PA}} = {\overrightarrow v _P} - {\overrightarrow v _A}$$$${\overrightarrow v _{PA}} = \left( {{v_2} + {v_1}} \right)\widehat i - 0$$$${\overrightarrow v _{PA}} = \left( {{v_2} + {v_1}} \right)\widehat i$$ So, observer $A$ will observe same velocity. | $${\overrightarrow v _P} = \left( {{v_2} + {v_1}} \right)\widehat i$$$${\overrightarrow v _B} = {v_1}\widehat i$$ So, $${\overrightarrow v _{PB}} = {\overrightarrow v _P} - {\overrightarrow v _B}$$$${\overrightarrow v _{PB}} = \left( {{v_2} + {v_1}} \right)\widehat i - {v_1}\widehat i$$$${\overrightarrow v _{PB}} = {v_2}\widehat i$$ So, $B$ will observe that passenger is going away from him with velocity |
| Velocity of $A$ $\left( {{{\overrightarrow v }_A}} \right)$ | $${\overrightarrow v _{AA}} = {\overrightarrow v _A} - {\overrightarrow v _A}$$$${\overrightarrow v _{AA}} = 0$$ | As, $${\overrightarrow v _A} = 0$$$${\overrightarrow v _B} = {v_1}\widehat i$$So, $${\overrightarrow v _{AB}} = {\overrightarrow v _A} - {\overrightarrow v _B}$$$${\overrightarrow v _{AB}} = 0 - {v_1}\widehat i$$$${\overrightarrow v _{AB}} = - {v_1}\widehat i$$ |
| Velocity of $B$ $\left( {{{\overrightarrow v }_B}} \right)$ | As, $${\overrightarrow v _A} = 0$$$${\overrightarrow v _B} = {v_1}\widehat i$$So, $${\overrightarrow v _{BA}} = {\overrightarrow v _B} - {\overrightarrow v _A}$$$${\overrightarrow v _{BA}} = {v_1}\widehat i-0$$$${\overrightarrow v _{BA}} = {v_1}\widehat i$$ | $${\overrightarrow v _{BB}} = {\overrightarrow v _B} - {\overrightarrow v _B}$$$${\overrightarrow v _{BB}} = 0$$ |
For better understanding, all the observation of both the observers is shown in the figure below,
