To analyze the situation involving two particles, A and B, connected by a light inextensible string, we need to consider the principles of momentum and the constraints imposed by the string. When particle B is projected with an initial speed \( u \) perpendicular to the line connecting A and B, we can determine the velocity of particle A just after B is projected.
Understanding the System
We have two particles, A and B, each with mass \( m \). The string connecting them has a length of \( 2l \), and initially, particle B is at a distance \( l \) from particle A. When B is projected with speed \( u \), it moves away from A, creating a scenario where we need to apply the conservation of momentum.
Applying Conservation of Momentum
Since the table is smooth, we can assume that there are no external horizontal forces acting on the system. Therefore, the horizontal momentum of the system must be conserved. Before B is projected, both particles are at rest, so the initial momentum of the system is zero.
When B is projected, it moves with velocity \( u \) in a direction perpendicular to the line connecting A and B. At this moment, we can denote the velocity of particle A as \( v_A \) and the velocity of particle B as \( u \). The conservation of momentum in the horizontal direction can be expressed as:
- Initial momentum = Final momentum
- 0 = m * v_A + m * u
Solving for the Velocity of A
From the conservation of momentum equation, we can isolate \( v_A \):
0 = m * v_A + m * u
=> m * v_A = -m * u
=> v_A = -u
This indicates that particle A moves in the opposite direction to particle B with a speed of \( u \) just after B is projected. The negative sign signifies that A's velocity is directed opposite to B's motion.
Visualizing the Motion
To visualize this, imagine particle A remaining stationary while particle B moves away. The tension in the string will cause A to accelerate in the opposite direction as B moves. However, immediately after B is projected, A's velocity is simply the result of momentum conservation, which we calculated to be \( -u \).
Key Takeaways
- The velocity of particle A just after B is projected is \( -u \).
- This result is derived from the conservation of momentum, which holds true in the absence of external forces.
- The direction of A's velocity is opposite to that of B's initial velocity.
In summary, the interaction between the two particles and the constraints of the string lead to a straightforward application of momentum conservation, allowing us to determine the velocity of particle A immediately after particle B is projected. This principle is fundamental in understanding systems involving connected objects and their motion in physics.
