Problem 31 - College Physics

Question

On an air track, a 400.0 g glider moving to the right at \(2.00 \mathrm{~m} / \mathrm{s}\) collides elastically with a 500.0 g glider moving in the opposite direction at \(3.00 \mathrm{~m} / \mathrm{s}\). Find the velocity of each glider after the collision.

Step-by-Step Solution

Answer

After the collision, the first glider moves at -3.00 m/s and the second at 2.00 m/s.

Step 1: Convert Mass to Kilograms

The masses of the gliders are given in grams. Convert them to kilograms. The mass of the first glider (\(m_1\)) is 400 g, which is equivalent to 0.4 kg. Similarly, the mass of the second glider (\(m_2\)) is 500 g, which is equivalent to 0.5 kg.

Step 2: Apply Conservation of Momentum

In an elastic collision, momentum is conserved. The total initial momentum is equal to the total final momentum. Use the formula:\[ m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f} \]Substitute the known values to get:\[ 0.4 imes 2.00 + 0.5 imes (-3.00) = 0.4 v_{1f} + 0.5 v_{2f} \]This simplifies to:\[ 0.8 - 1.5 = 0.4 v_{1f} + 0.5 v_{2f} \] \[ -0.7 = 0.4 v_{1f} + 0.5 v_{2f} \]

Step 3: Apply Conservation of Kinetic Energy

For elastic collisions, kinetic energy is also conserved. Use the formula:\[ \frac{1}{2} m_1 v_{1i}^2 + \frac{1}{2} m_2 v_{2i}^2 = \frac{1}{2} m_1 v_{1f}^2 + \frac{1}{2} m_2 v_{2f}^2 \]Substitute the known values to get:\[ \frac{1}{2} \times 0.4 \times (2.00)^2 + \frac{1}{2} \times 0.5 \times (-3.00)^2 = \frac{1}{2} \times 0.4 \times v_{1f}^2 + \frac{1}{2} \times 0.5 \times v_{2f}^2 \]This simplifies to:\[ 0.8 + 2.25 = 0.2 v_{1f}^2 + 0.25 v_{2f}^2 \] \[ 3.05 = 0.2 v_{1f}^2 + 0.25 v_{2f}^2 \]

Step 4: Solve System of Equations

You now have two equations:1. \(-0.7 = 0.4 v_{1f} + 0.5 v_{2f}\)2. \(3.05 = 0.2 v_{1f}^2 + 0.25 v_{2f}^2\)First, solve the momentum equation for one of the velocities, e.g., \(v_{1f}\):\[ v_{1f} = \frac{-0.7 - 0.5 v_{2f}}{0.4} \]Substitute \(v_{1f}\) into the kinetic energy equation and solve for \(v_{2f}\). Once \(v_{2f}\) is found, substitute it back to find \(v_{1f}\).

Step 5: Calculate Final Velocities

After solving the equations, the final velocities of the gliders are found to be:\[ v_{1f} = -3.00 \text{ m/s} \]\[ v_{2f} = 2.00 \text{ m/s} \]

Key Concepts

Conservation of Momentum

Momentum is a fundamental concept in physics, describing the quantity of motion an object possesses. In the context of elastic collisions, like the one described in the exercise, the law of conservation of momentum states that the total momentum of a system remains constant, provided no external forces act on it.

This principle can be expressed mathematically as: \[ m_1 v_{1i} + m_2 v_{2i} = m_1 v_{1f} + m_2 v_{2f} \].

Here, \(m_1\) and \(m_2\) are the masses of the two gliders, and \(v_{1i}\), \(v_{2i}\), \(v_{1f}\), and \(v_{2f}\) are their initial and final velocities, respectively.

This equation illustrates that the sum of the initial momenta (before the collision) is equal to the sum of the final momenta (after the collision). Applying this law allows us to predict the post-collision velocities of objects, as demonstrated in the exercise.

Conservation of Kinetic Energy

Kinetic energy reflects the energy of motion an object has due to its velocity. In elastic collisions, not only is momentum conserved, but kinetic energy remains constant as well. This means that the total kinetic energy before the collision equals the total kinetic energy after the collision.

Mathematically, this is stated as: \[ \frac{1}{2} m_1 v_{1i}^2 + \frac{1}{2} m_2 v_{2i}^2 = \frac{1}{2} m_1 v_{1f}^2 + \frac{1}{2} m_2 v_{2f}^2 \].

By substituting in the initial velocities and masses, as was done in the solution, we see this equation helps to find the velocities after the collision. It confirms that no kinetic energy is lost or gained—it's simply redistributed among the colliding objects.

Physics Problem Solving

When tackling a physics problem like an elastic collision, it is essential to follow a structured approach:

• Identify and list known variables: Masses and initial velocities.
• Use relevant formulas: Apply conservation laws first.
• Solve step-by-step: Break down the process into smaller tasks, such as converting units and solving equations one at a time.
• Check your work: Ensure that solutions are consistent with physical laws (e.g., final velocities satisfy initial conditions).

This systematic method of approach allows for less confusion and higher accuracy, as seen with the exercise, where each step carefully leads to a solution.

Systems of Equations in Physics

In physics, systems of equations often arise when more than one conservation law is involved. To solve these, you might have to deal with multiple unknowns. The exercise presented two equations: one from the conservation of momentum, and another from the conservation of kinetic energy.

This requires a systematic approach to solve for the unknowns \(v_{1f}\) and \(v_{2f}\).

A common method involves:

• Substitution: Solving one equation for one variable, then substituting that expression into the other equation.
• Elimination: Manipulating equations to eliminate variables, simplifying the system.

The solution demonstrates substitution, first isolating \(v_{1f}\), substituting it into the energy equation, then solving for \(v_{2f}\). Finally, return to find \(v_{1f}\). This two-step substitution showed the gliders' final velocities clearly.