To solve this problem, we need to apply the principles of conservation of momentum and energy. We have a small particle of mass \( m \) moving horizontally with a velocity \( u \) that strikes a triangular wedge of mass \( M \). After the collision, the particle moves up the incline of the wedge. Our goal is to find the velocity of the wedge immediately after the collision and the maximum height the particle can ascend on the wedge.
Step 1: Analyzing the Collision
Since the surface is frictionless, we can assume that the only forces acting on the system are those due to the collision. Before the collision, the momentum of the system is entirely due to the particle:
- Initial momentum of the particle: \( p_{\text{initial}} = mu \)
- Initial momentum of the wedge: \( p_{\text{wedge}} = 0 \) (since it is at rest)
The total initial momentum of the system is:
Initial momentum: \( p_{\text{total initial}} = mu + 0 = mu \)
Step 2: Conservation of Momentum
After the collision, let \( v \) be the velocity of the wedge and \( v_p \) be the velocity of the particle along the incline. The momentum of the system after the collision can be expressed as:
Final momentum: \( p_{\text{total final}} = Mv + mv_p \)
According to the conservation of momentum:
Equation 1: \( mu = Mv + mv_p \)
Step 3: Analyzing the Particle's Motion on the Wedge
As the particle moves up the incline, it will convert its kinetic energy into potential energy. The maximum height \( h \) that the particle can reach can be determined using energy conservation principles. The kinetic energy of the particle before it ascends is:
Kinetic Energy: \( KE = \frac{1}{2} mv^2 \)
When the particle reaches its maximum height, all of this kinetic energy will have converted into potential energy:
Potential Energy: \( PE = mgh \)
Setting the kinetic energy equal to the potential energy gives us:
Equation 2: \( \frac{1}{2} mv_p^2 = mgh \)
From this, we can solve for \( h \):
Height: \( h = \frac{v_p^2}{2g} \)
Step 4: Relating Velocities
To find the relationship between \( v_p \) and \( v \), we can use the geometry of the triangle. If \( \theta \) is the angle of the incline, then:
Relationship: \( v_p = v \sin(\theta) \)
Step 5: Solving the Equations
Substituting \( v_p \) into Equation 1 gives:
Equation 3: \( mu = Mv + m(v \sin(\theta)) \)
Rearranging this, we can express \( v \) in terms of \( u \), \( M \), and \( m \):
Equation 4: \( v(M + m \sin(\theta)) = mu \)
Thus, we find:
Velocity of the wedge: \( v = \frac{mu}{M + m \sin(\theta)} \)
Step 6: Finding Maximum Height
Now substituting \( v_p = v \sin(\theta) \) into Equation 2 gives:
Maximum height: \( h = \frac{(v \sin(\theta))^2}{2g} \)
Substituting \( v \) from Equation 4 into this expression will yield the maximum height the particle can ascend on the wedge.
In summary, by applying the principles of conservation of momentum and energy, we can determine both the velocity of the wedge immediately after the collision and the maximum height the particle can reach on the incline. This approach illustrates the interconnectedness of momentum and energy in a collision scenario.
