Understanding Relative Motion Made Easy

Relative motion in physics refers to the process of measuring the movement of one object with respect to another object, which may itself be moving or stationary. All motion is described relative to a chosen frame of reference, and there is no concept of absolute motion or absolute rest. Analyzing relative motion is fundamental for understanding problems involving multiple moving bodies and is widely applied in kinematics.

Definition of Relative Motion

Relative motion is defined as the motion of an object as observed from a specific reference frame, which itself may be at rest or in motion. The displacement, velocity, and acceleration of the object are always specified with respect to this reference frame.

In mathematical terms, if object A and object B have velocities $\vec{v}_A$ and $\vec{v}_B$ (relative to a ground or common reference), the velocity of A relative to B is given by: $\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$

The negative sign indicates that the observer on B sees A moving at a velocity that is the difference between their respective velocities. More details on velocity can be found at Relative Velocity Explained.

Reference Frame and Observers

A reference frame consists of a coordinate system and a clock from which measurements are made. It determines how motion is described. A reference frame can be inertial (not accelerating) or non-inertial (accelerating). Most classical mechanics problems consider inertial frames for simplicity.

The motion of an object may appear different to observers in different reference frames. For example, a passenger sitting in a moving train may appear to be at rest with respect to another passenger, but appears to be moving to an observer standing on the ground.

Concepts regarding frames of reference are explored further under Understanding Relativity.

Relative Velocity in One Dimension

In cases where two objects move along a straight line (one-dimensional motion), the relative velocity of A with respect to B is given by $\vec{v}_{AB} = \vec{v}_A - \vec{v}_B$. The sign and magnitude depend on the direction and the chosen positive axis.

When two objects move in the same direction, the magnitude of their relative velocity is the difference in their speeds. When they move in opposite directions, their speeds add algebraically due to opposite signs.

Relative Acceleration

Relative acceleration is defined as the time derivative of relative velocity. Mathematically, the acceleration of A relative to B is: $\vec{a}_{AB} = \vec{a}_A - \vec{a}_B$

Relative acceleration is used when the accelerations of the objects are not the same or when the reference frame itself is accelerating. Key principles in non-uniform accelerated motion are explained at Non-Uniform Accelerated Motion.

Equations for Relative Motion with Constant Acceleration

The standard kinematic equations apply to relative motion when acceleration is constant. If $u_{\text{rel}}$ is the initial relative velocity, $a_{\text{rel}}$ is the relative acceleration, and $s_{\text{rel}}$ is the relative displacement, the equations are as follows:

$v_{\text{rel}} = u_{\text{rel}} + a_{\text{rel}} t$

$s_{\text{rel}} = u_{\text{rel}} t + \dfrac{1}{2} a_{\text{rel}} t^2$

$v_{\text{rel}}^2 = u_{\text{rel}}^2 + 2 a_{\text{rel}} s_{\text{rel}}$

Relative Motion in Two Dimensions

When the motion occurs in two dimensions, relative velocities must be considered as vectors. If $\vec{v}_A$ and $\vec{v}_B$ are velocities of two objects, the relative velocity $\vec{v}_{AB}$ is calculated using vector subtraction.

Applications involve adding and subtracting vector quantities, and the concept is significant in analyzing problems like particle collisions and projectile motion. For further details, refer to Motion in 2D Dimensions.

Typical Relative Motion Examples

Consider two vehicles moving along a straight road. If car A moves at 80 km/h and car B moves at 60 km/h in the same direction, the velocity of A relative to B is $v_{AB} = 80 - 60 = 20$ km/h. If they move in opposite directions, $v_{AB} = 80 - (-60) = 140$ km/h.

In the case of two objects projected in different directions, their paths and velocities are resolved into components, and relative positions are calculated accordingly. Example problems enhance understanding by applying the vector approach.

Boat and River Problems: Relative Motion Application

In boat and river problems, the velocity of the boat with respect to water and the velocity of water with respect to ground are combined using vector addition to determine the boat's velocity relative to the ground. This method is directly applicable to similar problems involving wind and aeroplanes.

• Boat moving downstream: velocities add
• Boat moving upstream: velocities subtract

If a man swims with velocity $\vec{v}_{MR}$ relative to the river and the river flows with velocity $\vec{v}_R$, then the velocity relative to the ground is: $\vec{v}_{MG} = \vec{v}_{MR} + \vec{v}_R$ More on river crossing and vector resolution is available at Kinematics Overview.

Drift and Time to Cross for River Problems

When crossing a river, the time required is determined by the component of swimmer's velocity perpendicular to the river. If $d$ is the width of the river, and $v_{MR}\sin\theta$ is the perpendicular component, the time to cross is: $t = \dfrac{d}{v_{MR} \sin\theta}$

Drift is the downstream distance moved by the swimmer due to the river's flow. The minimum drift occurs when the swimmer takes a suitable angle with respect to the direction of river flow.

Relative Motion in Wind and Aeroplane Problems

Problems involving aeroplanes and wind are analogous to boat and river problems. The plane's velocity with respect to air and the air's velocity with respect to the ground are vectorially added to obtain the resultant or ground velocity.

• Velocity with respect to ground: sum of airspeed and wind velocity
• Course correction uses vector addition

Resolution of vectors is required to determine drift and resultant direction, similar to river problems.

Rain and Relative Motion Observation

The path of falling rain appears different to stationary and moving observers. If the rain falls vertically with velocity $\vec{v}_R$ and an observer moves horizontally with velocity $\vec{v}_M$, the rain appears to come at an angle. The apparent angle $\theta$ with the vertical is: $\tan\theta = \dfrac{v_M}{v_R}$

Velocity of Approach and Separation

The velocity of approach is the rate at which the distance between two bodies decreases. If two particles move with velocities $\vec{v}_A$ and $\vec{v}_B$ along the same line, the velocity of approach is $\lvert v_A - v_B \rvert$, when they move towards each other. If the distance increases, it is called velocity of separation.

Two-Dimensional Velocity of Approach

For two particles A and B moving at velocities $\vec{v}_1$ and $\vec{v}_2$ making angles $\theta_1$ and $\theta_2$ with the line joining them, the velocity of approach is:

$v_{\text{app}} = v_1 \cos\theta_1 + v_2 \cos\theta_2$

This value reflects the component of the relative velocity along the line joining the two objects.

Key Differences: Absolute vs Relative Quantities

For additional explanation on velocity measurement, visit Instantaneous Velocity Insights.