91Ó°ÊÓ

When dealing with collisions, the conservation of momentum is a fundamental principle that helps us understand the behavior of objects during a collision. Momentum is a measure of an object's motion and is the product of its mass and velocity. During a perfectly inelastic collision, like the one described in the problem, the two colliding objects stick together after the impact. This means their combined mass is now moving with a common velocity post-collision. The law of conservation of momentum states that the total momentum before the collision must equal the total momentum after the collision. This principle allows us to calculate the final velocity of the combined mass. In our scenario, two 10.0 kg masses collide; one is initially moving while the other is at rest. Applying the conservation of momentum:

• Initial momentum: 10.0 kg × 2.00 m/s + 10.0 kg × 0 m/s = 20.0 kg·m/s
• Final momentum of the combined mass: (10.0 kg + 10.0 kg) × v_f = 20.0 kg·m/s

Solving gives the final velocity, \(v_f = 1.00 \; \text{m/s}\), confirming the conservation of momentum in action.

A mass-spring system is a classical model used in physics to describe oscillatory motion, often idealized to ignore damping and other forces apart from the restoring force of the spring. In this system, a mass is attached to a spring with a known spring constant, represented by \(k\). Once an initial push or force is applied to this system, it will oscillate back and forth around an equilibrium position. The spring force, according to Hooke's Law, is proportional to the displacement but acts in the opposite direction. This restoring force causes the mass to accelerate back towards equilibrium. In our problem, after the collision, the two masses together act as a single mass of 20.0 kg, attached to the spring with a spring constant of 170.0 N/m. The interaction of the mass with the spring determines the oscillatory behavior of the system, characterized by its frequency, period, and amplitude of oscillation.

Angular frequency \(\omega\) is a key parameter in describing oscillatory motion, indicating how fast an object oscillates. In the context of a mass-spring system, it relates to how rapidly the system completes its cycles of movement. The formula to compute the angular frequency for a mass-spring system is: \[\omega = \sqrt{\frac{k}{m}}\]where \(k\) is the spring constant, and \(m\) is the total mass of the system. For our exercise, with a spring constant of 170.0 N/m and a combined mass of 20.0 kg, the angular frequency is calculated as: \[\omega = \sqrt{\frac{170}{20}} \approx 2.92 \; \text{rad/s}\]This value signifies the radians per second at which the system oscillates, providing insight into the system's dynamic properties.

The amplitude of oscillation describes the maximum displacement from the equilibrium position that the system achieves during its motion. It's a measure of how "big" the oscillation is. In a mass-spring system where kinetic and potential energy are exchanged, the amplitude can be determined by utilizing the conservation of energy principle. Immediately after the collision, all kinetic energy is converted into potential energy at the point of maximum displacement: \[\frac{1}{2} m v_f^2 = \frac{1}{2} k A^2\]Using this relation and plugging in values from our exercise (\(m = 20\; \text{kg}\), \(v_f = 1.00\; \text{m/s}\), \(k = 170\; \text{N/m}\)), we solve for the amplitude:\[A = \sqrt{\frac{20}{170}} \approx 0.344 \; \text{m}\]This means the system moves 0.344 meters away from equilibrium at its furthest point during oscillation. Understanding amplitude is crucial as it helps in predicting the extent of oscillation based on initial conditions.