Question

A moving particle has coordinates (5t + 3, 6t, 5)m in frame S at any time 't'. The frame s' is moving with velocity (3\(\hat{i}\)+4\(\hat{j}\))m/s with respect to the frame S. Velocity of particle in frame s' is:

• A. \( 2\hat{i} + 2\hat{j} \)
• B. \( 2\hat{i} - 2\hat{j} \)
• C. \( 3\hat{i} + 4\hat{j} \)
• D. \( 2\hat{i} \)

Hint:

Remember the subscription notation for relative velocity: \( \vec{v}_{A,B} \) means "velocity of A with respect to B". The transformation rule can be written as \( \vec{v}_{A,C} = \vec{v}_{A,B} + \vec{v}_{B,C} \). In this problem, let A = particle, B = frame S', C = frame S. We want \( \vec{v}_{p,S'} \). We know \( \vec{v}_{p,S} \) and \( \vec{v}_{S',S} \). The relation is \( \vec{v}_{p,S} = \vec{v}_{p,S'} + \vec{v}_{S',S} \), which gives the required formula.

Answer 1

The Correct Option is A

Step 1: Understanding the Concept:
This problem involves relative motion and requires the use of Galilean velocity transformation. We are given the position of a particle in one inertial frame (S) and the velocity of another inertial frame (S') relative to S. We need to find the velocity of the particle as observed from frame S'.
Step 2: Key Formula or Approach:
The Galilean transformation for velocities relates the velocity of a particle in frame S (\(\vec{v}_{p,S}\)), the velocity of the particle in frame S' (\(\vec{v}_{p,S'}\)), and the velocity of frame S' relative to S (\(\vec{v}_{S',S}\)):
\[ \vec{v}_{p,S} = \vec{v}_{p,S'} + \vec{v}_{S',S} \] To find the velocity of the particle in frame S', we rearrange this formula:
\[ \vec{v}_{p,S'} = \vec{v}_{p,S} - \vec{v}_{S',S} \] Step 3: Detailed Explanation:
1. Find the velocity of the particle in frame S (\(\vec{v}_{p,S}\)):
The position vector of the particle in frame S is given by:
\[ \vec{r}_{p,S}(t) = (5t + 3)\hat{i} + (6t)\hat{j} + (5)\hat{k} \] To find the velocity, we differentiate the position vector with respect to time \( t \):
\[ \vec{v}_{p,S} = \frac{d\vec{r}_{p,S}}{dt} = \frac{d}{dt}[(5t + 3)\hat{i} + (6t)\hat{j} + (5)\hat{k}] \] \[ \vec{v}_{p,S} = 5\hat{i} + 6\hat{j} + 0\hat{k} = (5\hat{i} + 6\hat{j}) \, \text{m/s} \] 2. Identify the velocity of frame S' relative to S (\(\vec{v}_{S',S}\)):
This is given in the problem:
\[ \vec{v}_{S',S} = (3\hat{i} + 4\hat{j}) \, \text{m/s} \] 3. Calculate the velocity of the particle in frame S' (\(\vec{v}_{p,S'}\)):
Using the transformation formula:
\[ \vec{v}_{p,S'} = \vec{v}_{p,S} - \vec{v}_{S',S} \] \[ \vec{v}_{p,S'} = (5\hat{i} + 6\hat{j}) - (3\hat{i} + 4\hat{j}) \] \[ \vec{v}_{p,S'} = (5-3)\hat{i} + (6-4)\hat{j} \] \[ \vec{v}_{p,S'} = (2\hat{i} + 2\hat{j}) \, \text{m/s} \] Step 4: Final Answer:
The velocity of the particle in frame S' is \( 2\hat{i} + 2\hat{j} \) m/s. This corresponds to option (A).