Problem 67 - College Physics

Question

A blue puck with mass \(0.0400 \mathrm{~kg}\), sliding with a velocity of magnitude \(0.200 \mathrm{~m} / \mathrm{s}\) on a frictionless, horizontal air table, makes a perfectly elastic, head-on collision with a red puck with mass \(m\) initially at rest. After the collision, the velocity of the blue puck is \(0.050 \mathrm{~m} / \mathrm{s}\) in the same direction as its initial velocity. Find (a) the velocity (magnitude and direction) of the red puck after the collision; and (b) the mass \(m\) of the red puck.

Step-by-Step Solution

Answer

(a) The red puck's velocity is 0.25 m/s in the same direction as the blue puck's initial velocity. (b) The mass of the red puck is 0.024 kg.

Step 1: Understand the Problem Requirements

We need to find two things: the velocity of the red puck after the collision and the mass of the red puck. The collision is perfectly elastic, so both momentum and kinetic energy are conserved.

Step 2: Set up the Conservation of Momentum Equation

In an elastic collision, the total linear momentum before and after the collision is conserved.Let the initial velocity of the red puck be 0 (since it is at rest), and the mass of the blue puck be 0.0400 kg with an initial velocity of 0.200 m/s.The equation for conservation of momentum is:\[ m_{blue} \cdot v_{blue, initial} + m_{red} \cdot v_{red, initial} = m_{blue} \cdot v_{blue, final} + m_{red} \cdot v_{red, final}. \]Substitute the known quantities:\[ 0.0400 \cdot 0.200 + m \cdot 0 = 0.0400 \cdot 0.050 + m \cdot v_{red, final}. \]Simplify the equation:\[ 0.008 = 0.002 + m \cdot v_{red, final}. \]

Step 3: Solve for the Velocity of the Red Puck

Rearrange the momentum equation to solve for the velocity of the red puck:\[ m \cdot v_{red, final} = 0.008 - 0.002 = 0.006. \]The expression for the velocity of the red puck is:\[ v_{red, final} = \frac{0.006}{m}. \]

Step 4: Set up the Conservation of Kinetic Energy Equation

For a perfectly elastic collision, the total kinetic energy is also conserved. Set up the equation for conservation of kinetic energy:\[ \frac{1}{2} m_{blue} \cdot v_{blue, initial}^2 + \frac{1}{2} m_{red} \cdot v_{red, initial}^2 = \frac{1}{2} m_{blue} \cdot v_{blue, final}^2 + \frac{1}{2} m_{red} \cdot v_{red, final}^2. \]Substitute the known values:\[ 0.5 \cdot 0.0400 \cdot (0.200)^2 + 0 = 0.5 \cdot 0.0400 \cdot (0.050)^2 + 0.5 \cdot m \cdot \left(\frac{0.006}{m}\right)^2. \]Simplify and solve for \(m\):\[ 0.0008 = 0.00005 + 0.5 \cdot \frac{0.006^2}{m}. \]

Step 5: Solve for the Mass of the Red Puck

Rearrange the kinetic energy equation to find the mass \(m\):\[ 0.0008 - 0.00005 = \frac{0.000018}{m}. \]\[ 0.00075 = \frac{0.000018}{m}. \]Thus, \( m = \frac{0.000018}{0.00075} \approx 0.024 \) kg.

Step 6: Calculate the Final Velocity of the Red Puck

With the mass \( m = 0.024 \) kg, substitute back into the expression we found for the velocity of the red puck:\[ v_{red, final} = \frac{0.006}{0.024} = 0.25 \text{ m/s}. \]The red puck moves in the direction of the blue puck’s initial velocity.

Key Concepts

Conservation of Momentum

In an elastic collision, one of the core principles is the conservation of momentum. This means that the total momentum of two objects before they collide is equal to the total momentum after the collision.
In this problem, the blue puck and the red puck make up a system where the momentum conservation principle applies.
Even though the red puck is at rest initially, its final velocity can be determined by applying this principle.

• The momentum formula is given by: \( p = m imes v \), where \( m \) is mass and \( v \) is velocity.
• For the blue and red pucks system, we set up the equation: \[ 0.0400 imes 0.200 + m imes 0 = 0.0400 imes 0.050 + m imes v_{red, final}. \]
• From this, we can find that the post-collision momentum must still sum up to the initial momentum of the blue puck since it starts the system moving.

Understanding and applying the conservation of momentum is crucial in solving physics problems involving collisions.
By simplifying the equation, we can reveal insights about the unknowns, such as the velocity of a colliding object.

Conservation of Kinetic Energy

Another key principle in elastic collisions is the conservation of kinetic energy. In such events, the total kinetic energy in the system doesn't change; it only transfers between the colliding bodies.
For our pucks, this means the energy remains constant before and after they collide.

• Kinetic energy is calculated using: \( KE = \frac{1}{2} m v^2 \).
• The equation for our system becomes: \[ 0.5 \times 0.0400 \times (0.200)^2 + 0 = 0.5 \times 0.0400 \times (0.050)^2 + 0.5 \times m \times \left(\frac{0.006}{m}\right)^2. \]
• By setting up and solving this kinetic energy equation, we can find the mass of the red puck by leveraging the total energy conservation.

This principle sets elastic collisions apart from inelastic collisions, where some kinetic energy is lost to other forms, such as heat.
In elastic collisions like ours, understanding how energy transfers lead to key solutions in physics problems.

Inelastic Collision

While this problem revolves around elastic collision, it's important to distinguish it from an inelastic collision.
In an inelastic collision, the objects stick together or deform, resulting in a loss of kinetic energy from the system. This energy doesn't dissipate into the environment as kinetic energy anymore, but instead might turn into heat or sound.

• Momentum is still conserved, but total kinetic energy is not.
• Computations for inelastic collisions depend on different equations that don't equate initial and final kinetic energies.

Although inelastic collisions are not the focus of this problem, understanding the differences helps in analyzing various collision scenarios.
It's critical for students to know when energy conservation applies to avoid errors in physics problem-solving contexts involving collisions.

Problem-Solving Strategies in Physics

Tackling physics problems effectively often involves a set of strategic steps.
Understanding the key principles, such as conservation laws, can significantly simplify complex problems like this elastic collision scenario.
Begin by thoroughly understanding the problem statement and identifying what is known and what needs to be found.

• Set up equations based on foundational laws, like conservation of momentum and energy.
• Simplify these equations step by step to find unknown quantities.
• Check the consistency of your solution with the principles applied, ensuring momentum and energy are appropriately conserved.
• If necessary, review definitions and distinctions, like those between elastic and inelastic collisions, to apply the right concepts.

These strategies not only help in physics exams but also in real-world applications.
They provide a framework for approaching a variety of problems using logic and physical principles effectively and consistently.