Problem 10 - College Physics

Question

On a frictionless air track, a \(0.150 \mathrm{~kg}\) glider moving at \(1.20 \mathrm{~m} / \mathrm{s}\) to the right collides with and sticks to a stationary \(0.250 \mathrm{~kg}\) glider. (a) What is the net momentum of this two-glider system before the collision? (b) What must be the net momentum of this system after the collision? Why? (c) Use your answers in parts (a) and (b) to find the speed of the gliders after the collision. (d) Is kinetic energy conserved during the collision?

Step-by-Step Solution

Answer

(a) 0.180 kg·m/s (b) 0.180 kg·m/s (c) 0.450 m/s (d) No, kinetic energy is not conserved.

Step 1: Calculate Initial Momentum

The initial momentum of the system can be determined using the formula for linear momentum, which is \( p = mv \), where \( m \) is the mass and \( v \) is the velocity. The initial momentum only comes from the moving glider, i.e., \( p_{ ext{initial}} = 0.150 \, \text{kg} \times 1.20 \, \text{m/s} = 0.180 \, \text{kg} \cdot \text{m/s} \).

Step 2: Determining Final Momentum

According to the law of conservation of momentum, the total momentum of a closed system remains constant if no external forces act on it. Hence, the net momentum after the collision is equal to the initial momentum. Thus, \( p_{ ext{final}} = 0.180 \, \text{kg} \cdot \text{m/s} \).

Step 3: Solve for Final Speed

To find the final speed of the combined gliders, use the conservation of momentum formula: \( m_1v_1 + m_2v_2 = (m_1 + m_2)v_f \) where \( v_f \) is the final velocity. Substitute the known values: \( 0.180 = (0.150 + 0.250)v_f \), solve to find \( v_f = \frac{0.180}{0.400} = 0.450 \, \text{m/s} \).

Step 4: Check Kinetic Energy Conservation

Kinetic energy before collision is \( KE_{ ext{initial}} = \frac{1}{2} m v^2 = \frac{1}{2} \times 0.150 \times (1.20)^2 = 0.108 \, \text{J} \). After collision, \( KE_{ ext{final}} = \frac{1}{2} \times 0.400 \times (0.450)^2 = 0.0405 \, \text{J} \). The kinetic energy is not conserved because \( 0.0405 \, \text{J} eq 0.108 \, \text{J} \). This is due to the inelastic nature of the collision.

Key Concepts

Inelastic Collision

In an inelastic collision, the objects involved collide and stick together, as seen with the gliders in our exercise. Unlike elastic collisions, inelastic collisions do not conserve kinetic energy. This is because some of the kinetic energy is transformed into other forms of energy, such as heat or sound, during the collision process. In our example, when the moving glider hits the stationary one, they stick together, forming a combined mass. This behavior confirms the collision is inelastic. While the kinetic energy isn't conserved, the momentum is still maintained before and after the collision.

Momentum

Momentum is a fundamental concept in physics, representing the quantity of motion an object possesses. It is calculated using the formula \( p = mv \), where \( p \) is the momentum, \( m \) is the mass, and \( v \) is the velocity. In the exercise with the gliders, the momentum before and after the collision remains the same due to the conservation principle. Before the collision, only the moving glider contributes to the system's overall momentum. After the collision, the combined mass of the gliders means the system now moves at a reduced velocity but with the same total momentum as initially. This illustrates how momentum takes into account both mass and speed, maintaining balance in a frictionless scenario.

Kinetic Energy

Kinetic energy is the energy that an object possesses due to its motion and is given by the formula \( KE = \frac{1}{2} mv^2 \). In our context, the kinetic energy of the gliders is examined before and after the collision. Before the collision, only the moving glider has kinetic energy, calculated using its mass and speed. However, after the collision, despite both gliders moving together, their combined kinetic energy is less. This loss in kinetic energy is indicative of an inelastic collision, where energy is not conserved. The difference highlights how not all energy remains as motion energy post-collision, as some are transferred to other forms.

Frictionless Surface

A frictionless surface is an idealized concept where no opposing forces slow down or alter the motion of objects. In the given exercise, the frictionless air track ensures that the only forces at play are due to the collision itself. This absence of friction allows us to focus solely on the momentum and energy changes without external interference. The gliders can thus freely translate all momentum and most energy dynamics from the collision into motion. Without resistance, the principles of momentum conservation and energy transformation become clearer and easier to calculate, simplifying the problem-solving process.