Momentum - Problem Solving 1D

Linear momentum is an important classical concept for describing the (change in) translational motion of an object. It gets defined by \( \overrightarrow{p} = m\cdot \overrightarrow{v} \) .

Linear momentum can be used to rephrase Newton's second law. If we assume mass is constant, we get \[ \overrightarrow{F} = \dfrac{d \overrightarrow{p}}{dt} = m \cdot \overrightarrow{a} .\] When mass is not constant, we can use the first equation to describe for example, rocket motion.

Although velocity and mass themselves can be used to describe the motion of an object and its changes, linear momentum gives us a different approach, which seems to come in very handy when dealing with collisions. It is so helpful that it survived the quantum revolution, where the concept of force was given up. %

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Conservation of momentum is seen by physicists as fundamental.

It is used in solving a variety of problems and especially in collisions. We can write it down in an equation \( \overrightarrow{p_{i}} = \overrightarrow{p_{f}} \) .

A baseball player with a mass of \(80.0\text{ kg}\) throws a \(145\text{ g}\) baseball with a speed of \(160 \text{ km/h}\) towards the batter.

What is the recoil speed of the batter? (Not yet finished.)

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Before the throw, neither the batter nor the baseball has speed. Therefore, \( \overrightarrow{p_{i}} = \overrightarrow{0} \).
After the throw, with index 1 for the player and 2 for the ball, we get \( \overrightarrow{p_{f}} = m_{1}\cdot \overrightarrow{v_{1}} + m_{2}\cdot \overrightarrow{v_{2}} \).

Using a one dimensional line, pointing in the direction of batter, and using conservation of momentum, we get \[ m_{1}\cdot v_{1} + m_{2}\cdot v_{2} = 0. \ _\square \]