Speed And Velocity

We are given two velocity equations for particles P and Q:

The condition is that both particles move with the same velocity at some time , so:

Now, let's solve for :

Move the terms involving to one side:

Factor out :

Finally, solve for :

Conclusion:

The time at which the velocities of particles P and Q are equal is given by:

Step 1: Analyze the Velocity-Time Graph 

The slope of the velocity-time (v-t) graph represents acceleration. From the given plot:

The velocity increases linearly with time, implying constant acceleration.

Step 2: Match with the Acceleration-Time Graph

Since the acceleration is constant, the acceleration-time (a-t) graph must be a horizontal straight line.

Step 3: Conclusion

The correct acceleration-time graph is option (3)

Step 1: Understand Rolling Motion

In rolling motion, a point on the rim of the wheel has both rotational and translational motion. The velocities due to both these motions add vectorially.

Step 2: Analyze the Velocity at Points P and Q

Point P is at the top of the wheel. Its velocity is the sum of the linear speed (v) of the wheel and the rotational speed (v).

Point Q is at the bottom of the wheel, where the linear speed (v) and rotational speed (-v) cancel out.

Step 3: Calculate Velocities

Velocity at P:

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Step 4: Conclusion

Point P moves faster than Point Q, and Point Q has zero velocity.