A particle C can move along a slide placed in a vertical plane. Points A and B are at the same horizontal level. Determine with respect to B the point where the velocity of C is zero for the first time in the three following cases:

a) C is released from A without initial velocity (Neglect friction).
b) C is launched downward from A along the slide (Neglect Friction).
c) C is released from A without initial velocity but is subjected to a frictional force.

To determine the point where the velocity of particle C is zero with respect to point B, we need to analyze the motion of the particle in each case.

a) When C is released from A without initial velocity and neglecting friction, the only force acting on it is gravity. The motion of the particle will be solely determined by the force of gravity. Since point B is at the same level as point A, the particle will accelerate downward due to gravity and eventually pass point B. At some point after passing B, the particle will start moving upward, and its velocity will become zero when it reaches its highest point in its trajectory, which is above point B.

b) When C is launched downward from A along the slide without initial velocity and neglecting friction, again, the only force acting on it is gravity. However, because the particle is launched downward, it will start with an initial velocity in the downward direction. As it moves along the slide, the force of gravity will accelerate it further downward, increasing its velocity. Eventually, the particle will reach its maximum velocity when it passes point B. After passing B, the force of gravity will slow down the particle, causing its velocity to decrease. The particle will come to a stop when its velocity becomes zero at some point below point B. The exact location where its velocity becomes zero will depend on the initial launch velocity and the angle of the slide.

c) When C is released from A without initial velocity but is subjected to a frictional force, we need to consider the effects of friction. Friction will oppose the motion of the particle, causing it to decelerate. As the particle moves along the slide, the frictional force will act opposite to its direction of motion, slowing it down. The particle will eventually come to a stop when the frictional force is equal in magnitude and opposite in direction to the force of gravity. At this point, the velocity of the particle will be zero. The exact location where the velocity becomes zero will depend on the magnitude of the frictional force and the angle of the slide.

In each case, calculating the exact location where the velocity of particle C becomes zero requires a detailed analysis of the forces involved and the motion of the particle using equations of motion and principles of mechanics.

a) In this case, since the particle is released from A without any initial velocity, it will start accelerating due to the force of gravity. As it moves downward along the slide, the velocity will increase. However, at some point, the particle will reach its maximum potential energy and start losing its kinetic energy. This occurs when the velocity becomes zero for the first time.

To determine the point where the velocity is zero, we need to analyze the forces at play. The only force acting on the particle is the force of gravity, which is always directed downward. As the particle moves along the slide, the force of gravity can be resolved into two components: one acting along the slide (parallel to the surface) and one acting perpendicular to the surface.

Since the particle is released without any initial velocity, the force along the slide will cause the particle to accelerate. However, since we want to find the point where the velocity is zero, we need to consider the opposite force that counteracts the force of gravity along the slide. This force should be equal in magnitude and opposite in direction to the component of the force of gravity along the slide.

At the point where the velocity is zero, these two forces will be equal and opposite, resulting in a net force of zero. This means that the component of the force of gravity along the slide will be equal to the force that counteracts it.

b) In this case, the particle is launched downward from point A along the slide. As it moves downward, the force of gravity will act on it, causing it to accelerate. The velocity will increase until the particle reaches its maximum potential energy and starts losing kinetic energy. The velocity will become zero at the point where the net force acting on the particle is zero.

Since the particle is launched downward, there is an initial velocity and it is moving in the opposite direction of the acceleration due to gravity. Therefore, at some point, there will be a balance between the initial velocity and the acceleration due to gravity. This balance will cause the velocity to become zero.

To determine the point where the velocity is zero, we need to find the position where the acceleration due to gravity equals the initial velocity. At this point, the net force acting on the particle will be zero, resulting in a zero velocity.

c) In this case, the particle is released from A without any initial velocity but is subjected to a frictional force. The force of friction opposes the motion of the particle, acting in the opposite direction to its velocity.

As the particle moves downward along the slide, the force of gravity will act on it, causing it to accelerate. However, the force of friction will also act on the particle, opposing its motion. Initially, the force of friction will counteract the force of gravity and the particle will start decelerating.

The point where the velocity is zero will be the position where the acceleration due to gravity equals the deceleration due to friction. At this point, the net force acting on the particle will be zero, resulting in a zero velocity.