A \(950-\mathrm{kg}\) compact car is moving with velocity \(\vec{v}_{1}=32 \hat{\imath}+\) \(17 \hat{\jmath} \mathrm{m} / \mathrm{s} .\) It skids on a frictionless icy patch and collides with a \(450-\mathrm{kg}\) hay wagon with velocity \(\vec{v}_{2}=12 \hat{\imath}+14 \hat{\jmath} \mathrm{m} / \mathrm{s} .\) If the two stay together, what's their velocity?

Linear momentum is a measure of an object's motion, quantitatively defined as the product of its mass and velocity. It's a vector quantity, meaning it has both magnitude and direction, portrayed by the equation: \[\begin{equation} \vec{P} = m \vec{v} \end{equation}\]
In the context of the exercise, the momentum of two vehicles before the collision is crucial to understanding the afterward scenario. Momentum is conserved in any closed system without external forces, meaning the total momentum before an event like a collision must equal the total momentum after the event.

• For the compact car: \[\begin{equation}\vec{P}_1 = m_1 \vec{v}_1 = 950(32 \hat{i} + 17 \hat{j}) \end{equation}\]
• For the hay wagon: \[\begin{equation}\vec{P}_2 = m_2 \vec{v}_2 = 450(12 \hat{i} + 14 \hat{j}) \end{equation}\]

The calculation of individual momentums helps set the stage for analyzing the outcome of the collision.

Collision physics examines the interaction between two or more objects when they come into contact and exert forces on each other. During a collision, energy and momentum are exchanged, but in a perfectly inelastic collision—as depicted in our exercise—the objects stick together and move with the same velocity afterward.

Perfectly Inelastic Collision

When two objects collide and move together, as in the given scenario, we deal with a perfectly inelastic collision. Such collisions are characterized by the maximum possible loss of kinetic energy consistent with the law of conservation of momentum.The total initial momentum, calculated as the sum of the momentums of both objects before impact, must equal the total final momentum, which translates into a combined entity post-collision:\[\begin{equation} \vec{P}_{total-final} = \vec{P}_{total-initial} \end{equation}\]Providing students with the understanding of the types of collisions and their outcomes enhances comprehension of the final velocity after such events.

Understanding the relationship between momentum and velocity is fundamental to solving problems involving collisions. The momentum of an object is directly proportional to its velocity, with mass being the constant of proportionality.

Resolving for Velocity

After a collision, to find the new velocity of the combined mass, we rework the momentum formula \[\begin{equation} \vec{P} = m \vec{v} \end{equation}\]
and isolate the velocity component:\[\begin{equation} \vec{v} = \frac{\vec{P}}{m} \end{equation}\]For the combined mass of the car and hay wagon, we apply this formula to find their shared velocity post-collision. This illustrates the direct momentum-velocity relationship:

• Calculate combined mass: \[\begin{equation} m_{total} = m_1 + m_2 \end{equation}\]
• Use the conserved total momentum: \[\begin{equation} \vec{P}_{total-final} = 35800 \hat{i} + 22450 \hat{j} \end{equation}\]
• Find the final velocity: \[\begin{equation} \vec{v}_{final} = \frac{\vec{P}_{total-final}}{m_{total}} \end{equation}\]

By relating momentum to velocity through this lens, students can better visualize how objects' speeds and directions change after collisions.