A cannon of mass m slides down a smooth inclined
plane forming the angle a(alpha) with the horizontal. At the moment when
the velocity of the cannon reaches v, it
fires a shell in a horizontal direction
with the result that the cannon stops
and the shell carries away the momentum
p. Suppose that the firing duration
is equal to t. What is the reaction force
R of the inclined plane averaged over
the time t?

To determine the average reaction force \( R \) of the inclined plane while the cannon fires the shell, we need to analyze the situation using the principles of momentum and forces acting on the cannon. Let's break it down step by step.

Understanding the System

We have a cannon of mass \( m \) sliding down an inclined plane at an angle \( \alpha \) with an initial velocity \( v \). When the cannon fires a shell horizontally, it experiences a change in momentum, and we need to find the average reaction force during the firing duration \( t \).

Momentum Before and After Firing

Before the cannon fires the shell, its momentum \( p_{\text{initial}} \) can be expressed as:

• \( p_{\text{initial}} = mv \)

When the cannon fires the shell, it stops, and the shell carries away the momentum. Let’s denote the mass of the shell as \( m_s \) and its velocity as \( v_s \). The momentum of the shell after firing is:

• \( p_{\text{shell}} = m_s v_s \)

Since the cannon comes to a stop, the momentum conservation principle gives us:

• \( mv = m_s v_s \)

Calculating the Reaction Force

During the firing process, the cannon experiences a change in momentum over the time \( t \). The average force \( F \) exerted by the shell on the cannon can be calculated using the impulse-momentum theorem:

• \( F \cdot t = \Delta p \)

Here, \( \Delta p \) is the change in momentum of the cannon, which is equal to \( mv \) (since it goes from \( mv \) to \( 0 \)). Therefore:

• \( F \cdot t = mv \)
• \( F = \frac{mv}{t} \)

Considering Forces on the Cannon

While the cannon is on the inclined plane, it experiences two main forces: the gravitational force acting downwards and the normal reaction force \( R \) from the inclined plane. The gravitational force can be resolved into two components:

• Perpendicular to the incline: \( mg \cos(\alpha) \)
• Parallel to the incline: \( mg \sin(\alpha) \)

During the firing, the cannon also experiences an upward force due to the reaction force \( R \) and the downward force due to gravity. The net force acting on the cannon in the vertical direction must equal the change in momentum due to the firing force:

• \( R - mg \cos(\alpha) = -F \)

Substituting \( F \) from our earlier calculation:

• \( R - mg \cos(\alpha) = -\frac{mv}{t} \)

Final Expression for the Reaction Force

Rearranging the equation gives us:

• \( R = mg \cos(\alpha) - \frac{mv}{t} \)

This expression provides the average reaction force \( R \) of the inclined plane over the time \( t \) while the cannon fires the shell. It accounts for both the gravitational force acting on the cannon and the force exerted due to the firing of the shell.

Summary

In summary, the average reaction force \( R \) of the inclined plane while the cannon fires the shell can be expressed as:

• \( R = mg \cos(\alpha) - \frac{mv}{t} \)

This formula highlights the interplay between gravitational forces and the momentum change due to the firing of the shell, providing a comprehensive understanding of the dynamics involved in this scenario.