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define-relative-velocity-in-1d

🚀 In kinematics, relative velocity refers to the velocity of an object as observed from a particular reference frame, typically another moving object. In one-dimensional motion, relative velocity can be understood by considering two objects moving along a straight line. The relative velocity of one object with respect to another is calculated by subtracting the velocity of the second object from the velocity of the first. This concept is crucial in analyzing situations where multiple objects are in motion, such as vehicles on a road or particles in a physics experiment.

Theory Explanation

Understanding Relative Velocity

Relative velocity is defined as the velocity of one object as observed from another object. In one dimension, if we have two objects A and B, moving along the same line, the relative velocity of A with respect to B is given by the formula: \( v_{AB} = v_A - v_B \). Here, \( v_A \) is the velocity of object A and \( v_B \) is the velocity of object B. This formula shows how fast object A is moving compared to object B, which helps in understanding their motion relative to each other.

\[ v_{AB} = v_A - v_B \]
Application of Relative Velocity

To apply the concept of relative velocity, consider the directions of motion. If both objects are moving in the same direction, the relative velocity will be the difference of their speeds. If they are moving in opposite directions, the relative velocity will be the sum of their speeds. This application is essential in problems involving collisions or when analyzing the motion of objects in a shared environment.

Sign Conventions in Relative Velocity

When calculating relative velocity, it is important to establish a sign convention. Typically, one direction is considered positive, and the opposite direction is negative. This helps in accurately determining the relative velocity. For example, if object A moves to the right (positive direction) and object B moves to the left (negative direction), the velocities would be assigned accordingly, ensuring that calculations reflect the actual motion of the objects.

Key Points

  • 🎯 Relative velocity is the velocity of one object as observed from another object.
  • 🎯 In one dimension, it is calculated using the formula: v_AB = v_A - v_B.
  • 🎯 Sign conventions are crucial for determining the correct relative velocities.
  • 🎯 Relative velocity can be positive or negative depending on the direction of motion.
  • 🎯 Understanding relative velocity helps in solving problems involving multiple moving objects.

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Examples:💡

Two cars are moving along a straight road. Car A is moving at 60 km/h and Car B is moving at 40 km/h in the same direction. Calculate the relative velocity of Car A with respect to Car B.

Solution:

Step 1: Identify the velocities of both cars: v_A = 60 km/h and v_B = 40 km/h.

Step 2: Use the relative velocity formula: v_AB = v_A - v_B = 60 km/h - 40 km/h.

\[ v_{AB} = 60 - 40 = 20 km/h \]

Step 3: The relative velocity of Car A with respect to Car B is 20 km/h in the direction of Car A's motion.

A train is moving east at 80 km/h and a cyclist is moving west at 20 km/h. Find the relative velocity of the train with respect to the cyclist.

Solution:

Step 1: Assign the velocities: v_train = 80 km/h (east) and v_cyclist = -20 km/h (west, negative direction).

Step 2: Calculate the relative velocity using the formula: v_relative = v_train - v_cyclist = 80 km/h - (-20 km/h).

\[ v_{relative} = 80 - (-20) = 80 + 20 = 100 km/h \]

Step 3: The relative velocity of the train with respect to the cyclist is 100 km/h towards the east.

Common Mistakes

  • Mistake: Students often forget to consider the direction of motion when calculating relative velocity, leading to incorrect signs in their calculations.

    Correction: Always establish a clear sign convention for the directions of motion before performing calculations.

  • Mistake: Confusing the relative velocity formula by adding velocities instead of subtracting them when both objects are moving in the same direction.

    Correction: Remember that relative velocity is calculated by subtracting the velocity of the second object from the first: v_AB = v_A - v_B.

  • Mistake: Neglecting to account for the case when objects are moving in opposite directions, which requires adding their speeds instead of subtracting.

    Correction: If objects are moving in opposite directions, use the formula v_AB = v_A + |v_B| to find the relative velocity.

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