How Long Does the Shot Last in a Sliding Cannon Scenario?

[utkarshakash]

Homework Statement

A cannon of mass M starts sliding freely down a smooth inclined plane at an angle α to the horizontal. After the cannon covered the distance l, a shot was fired, the shell leaving the cannon in the horizontal direction with a momentum p. As a consequence, the cannon stopped. Assuming the mass of the shell to be negligible, as compared to that of the cannon, determine the duration of the shot.

Homework Equations

Kinematics Equations
Conservation of Momentum

The Attempt at a Solution

Acceleration of cannon along incline = $gsinα$
Distance covered = l
Initial Velocity = 0
So final velocity of the cannon (v) = $\sqrt{2glsinα}$
If I take X-axis along the incline and Y-axis perpendicular to it
$\vec{p}_{i}=Mvcosα$
$\vec{p}_{f}=0$ (Since cannon stops)
Since I have to find duration of shot so that means I have to find the time interval during which the momentum of the cannon changed. Am I right with this assumption or does that means something else? If I'm right then how do I find it?

 

[anathema]

Thank you for your question. It seems like you have a good understanding of the initial conditions and the equations involved in this problem. To answer your question, yes, you are correct in assuming that you need to find the time interval during which the momentum of the cannon changes. This is because the duration of the shot is directly related to the change in momentum of the cannon.

To find the duration of the shot, you can use the conservation of momentum equation: \vec{p}_{i}=\vec{p}_{f}. In this case, the initial momentum of the cannon is Mvcosα and the final momentum is 0, so the equation becomes Mvcosα=0. This means that the change in momentum is Mvcosα. To find the duration of the shot, you can use the kinematics equation v=v_{0}+at, where v is the final velocity, v_{0} is the initial velocity, a is the acceleration, and t is the time. In this case, the final velocity is 0, the initial velocity is \sqrt{2glsinα}, and the acceleration is gsinα. Plug in these values and solve for t to find the duration of the shot.

I hope this helps. Let me know if you have any further questions.