Mastering Relative Motion: A Step-by-Step Guide

by | Feb 9, 2025 | Guides | 0 comments

Mastering Relative Motion: A Step-by-Step Guide

1. Topic Overview

Relative motion refers to how the motion of an object appears from different frames of reference. Understanding relative motion is crucial in physics and everyday life, such as when determining the speed of a moving train from inside another train, understanding airspeed for airplanes, or navigating river currents.

Real-Life Applications:

  • Motion of cars as seen from different vehicles
  • Airplane velocity relative to the wind
  • Boats moving in a river with a current
  • Space travel and satellite motion

2. Key Concepts

  • Frame of Reference: A coordinate system from which motion is observed.
  • Relative Velocity: The velocity of one object as observed from another moving object.
  • Vector Addition: Used to combine velocities from different reference frames.

Key Equations:

  1. Relative Velocity in One Dimension:
    • v_{A/B} = v_A - v_B
    • The velocity of object A relative to object B is found by subtracting B’s velocity from A’s velocity.
  2. Relative Velocity in Two Dimensions:
    • \mathbf{v_{A/B}} = \mathbf{v_A} - \mathbf{v_B}
    • For two-dimensional motion, use vector subtraction.
  3. Velocity of an Object Relative to the Ground:
    • \mathbf{v_{O/G}} = \mathbf{v_{O/M}} + \mathbf{v_{M/G}}
    • Where \mathbf{v_{O/G}} is the velocity of the object relative to the ground, \mathbf{v_{O/M}} is the velocity relative to the medium (e.g., air or water), and \mathbf{v_{M/G}} is the velocity of the medium relative to the ground.

Diagram Example:

(Illustration of two moving objects with different frames of reference, showing velocity vectors.)

3. Step-by-Step Explanation

  1. Choose a Reference Frame:
    • Decide from which object or observer the motion is being measured.
  2. Identify Given Velocities:
    • Determine velocities relative to different objects.
  3. Use Vector Addition/Subtraction:
    • In one dimension, use simple subtraction.
    • In two dimensions, use vector components:
      • v_x = v \cos\theta
      • v_y = v \sin\theta
  4. Interpret Results:
    • If the result is positive, the object moves in the assumed direction.
    • If negative, the direction is opposite.

4. Worked Example

Problem:
A boat is moving at 4.50 m/s relative to a river, which flows at 2.30 m/s eastward. The boat heads north. Find the boat’s velocity relative to an observer on the riverbank.

Step 1: Define Reference Frames

  • Let the river be the medium.
  • Boat’s velocity relative to the river:
    • \mathbf{v_{B/R}} = (0, 4.50) \text{ m/s}
  • River’s velocity relative to the ground:
    • \mathbf{v_{R/G}} = (2.30, 0) \text{ m/s}

Step 2: Use Vector Addition

\mathbf{v_{B/G}} = \mathbf{v_{B/R}} + \mathbf{v_{R/G}}

\mathbf{v_{B/G}} = (0, 4.50) + (2.30, 0)

\mathbf{v_{B/G}} = (2.30, 4.50) \text{ m/s}

Step 3: Find Magnitude and Direction

Magnitude:
v = \sqrt{(2.30)^2 + (4.50)^2}

v = \sqrt{5.29 + 20.3}

v = \sqrt{25.6}

v = 5.06 \text{ m/s}

Direction:
\theta = \tan^{-1} \left( \frac{4.50}{2.30} \right)

\theta = \tan^{-1} (1.96)

\theta = 62.2^\circ (north-east)

Final Answer:

The boat moves at 5.06 m/s at 62.2° northeast relative to the ground.

💡 Tip: Always break vectors into components for 2D problems!

5. Practice Problems

  1. A plane is flying at 250 m/s north, while a wind of 60 m/s blows westward. Find the velocity of the plane relative to the ground.
  2. Two cars travel along a straight road. Car A moves at 30 m/s eastward, while Car B moves at 20 m/s westward. What is the velocity of Car A relative to Car B?
  3. A swimmer moves at 1.80 m/s in still water. If a river flows at 0.900 m/s, what is the swimmer’s velocity relative to the riverbank if they swim directly across the river?

6. Quick Summary & Key Takeaways

✅ Relative velocity depends on the observer’s frame of reference.
✅ Use vector addition when dealing with two-dimensional motion.
✅ For moving mediums (e.g., air, water), use:

  • \mathbf{v_{O/G}} = \mathbf{v_{O/M}} + \mathbf{v_{M/G}}
    ✅ Always break vectors into components for easier calculation.

7. Self-Check Quiz

Q1: A bus is moving at 15.0 m/s eastward, while a passenger inside walks 2.50 m/s westward. What is the passenger’s velocity relative to the ground?

Q2: A jet flies at 600 m/s due north while the wind blows at 50 m/s westward. Find the jet’s velocity relative to the ground.

Q3: A river flows at 3.00 m/s east, and a boat moves 5.00 m/s north relative to the river. What is the boat’s speed and direction relative to the shore?

(Check your answers using vector addition!)

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