9.6
Conservation of Momentum: Problem Solving
Solving problems using the conservation of momentum requires four basic steps:
- Identify a closed system, where the total mass is constant, and no net…
Consider a 50 kilogram stationary cannon. A shell of mass 5 kilograms is fired from the cannon which travels with a velocity of 400 meters per second. What will be the velocity of the cannon after firing, ignoring frictional forces?
The mass and velocity of the shell and the mass of the cannon are the known quantities. The velocity of the cannon is the unknown quantity. Ignoring frictional forces, the cannon and the shell will form a closed system.
The positive x-axis is assumed in the direction of the shot to define the velocity vectors. Since the cannon and shell are stationary, the momentum before the interaction is zero.
After firing the shell, the final momentum will be equal to the sum of the cannon's momentum and the shell's momentum.
Lastly, equate the momentum before and after the interaction to obtain the unknown quantity. Here, the velocity of the cannon after firing is resolved by substituting the known quantities in the equation and it comes out to be minus 40 meters per second.
The negative sign indicates that the cannon recoils in the direction opposite to the direction of the shot.
View the full transcript and gain access to JoVE Core videos
Q1: What are the four basic steps for solving conservation of momentum problems?
The four steps are: identify a closed system where total mass is constant and no net external force acts on it; write an expression for total momentum before the interaction; write an expression for total momentum after the interaction; and equate these two expressions to find the unknown quantity. These steps apply systematically to any momentum conservation problem.
Q2: Why is a closed system important when applying conservation of momentum?
A closed system ensures that total mass remains constant and no net external force acts on the system. Internal forces between objects do not change the system's total momentum. For example, when two carts collide and stick together, their weights are canceled by normal forces from the track, making it a valid closed system for momentum analysis.
Q3: How do you determine the direction of recoil in a momentum problem?
Define a positive direction along the x-axis before solving. If the calculated velocity is negative, the object recoils opposite to the positive direction. For instance, when a cannon fires a shell in the positive direction, the cannon's negative velocity indicates it recoils backward, opposite to the shell's motion.
Q4: What does it mean when initial momentum equals zero in a collision problem?
When both objects are stationary before interaction, their combined initial momentum is zero. By conservation of momentum, the final momentum must also equal zero. This means the momenta of the two objects after collision must be equal in magnitude but opposite in direction, ensuring the system's total momentum remains zero.
Q5: How do you calculate the final velocity of two objects that stick together after collision?
Use the conservation of momentum equation: initial momentum equals final momentum. For two carts sticking together, m₁v₁ + m₂v₂ = (m₁ + m₂)vf. Substitute known masses and velocities, then solve for the final velocity vf. This method works for any perfectly inelastic collision where objects combine into one.
Q6: What role do internal forces play in momentum conservation?
Internal forces between objects within a closed system do not affect the system's total momentum. When two carts collide, the magnetic forces they exert on each other are internal forces. These forces change individual object momenta but cancel out when calculating total system momentum, preserving overall momentum conservation.
Q7: How do you apply momentum conservation to problems involving multiple dimensions?
Momentum is conserved independently in each dimension. Define separate x and y axes, write momentum equations for each direction, and solve them independently. When analyzing collisions in multiple dimensions, apply the same conservation principles to both horizontal and vertical components to find the complete final velocity vector.