Question

Momentum for a system can be conserved in one direction while not being conserved in another. What is the angle between the directions? Givean example.

Short answer

The angle between the directions must be$90°$

Step-by-step solution

Definition of Momentum

The product of a particle's mass and velocity is called momentum. The term "momentum" refers to a quantity that has both a magnitude and a direction. The temporal rate of change in momentum is equal to the force applied on the particle, according to Isaac Newton's second law of motion.

The momentum of an object, $p=mv$, is a vector quantity. That is it has both magnitude and direction.

In the vector form, we can write the momentum as,$\overrightarrow{p}=m\overrightarrow{v}$

A vector can be resolved into two components, one is along the x-direction and the other is along the y-direction. The below figure represents the vector diagram of $\overrightarrow{p}$

where$\overrightarrow{p}=p_y\sin \theta$is the y-component of the$\overrightarrow{p}$ along the y-direction and$\overrightarrow{p}=p_x\sin \theta$ is the x-component of$\overrightarrow{p}$ along y-direction. Where $p_x$and $p_y$ are the magnitudes of $\overrightarrow{p}$ along the x and y-direction.

Explains the answer with an example

If the net external force acts along anyone of these x and y-directions is zero, then the momentum is conserved along only that direction. Thus the angle between these directions must be 90° as shown in the diagram given above.

One example of conservation of momentum is during projectile motion, if the air resistance is negligible; momentum is conserved in the horizontal(x-direction) direction because net horizontal forces acting along this direction is zero.