Problem 34 - University Physics with Modern Physics

Question

Two fun-loving otters are sliding toward each other on a muddy (and hence frictionless) horizontal surface. One of them, of mass \(7.50 \mathrm{kg},\) is sliding to the left at \(5.00 \mathrm{m} / \mathrm{s},\) while the other, of mass \(5.75 \mathrm{kg},\) is slipping to the right at 6.00 \(\mathrm{m} / \mathrm{s} .\) They hold fast to each other after they collide. (a) Find the magnitude and direction of the velocity of these free-spirited otters right after they collide. (b) How much mechanical energy dissipates during this play?

Step-by-Step Solution

Answer

(a) Velocity: \(-0.226 \text{ m/s}\) left. (b) Energy dissipated: \(196.91 \text{ J}\).

Step 1: Understand the Problem

We have two otters sliding toward each other on a frictionless surface. Their masses and velocities are given: Otter 1 has a mass of \( 7.50 \, \text{kg} \) and velocity \( 5.00 \, \text{m/s} \) to the left, and Otter 2 has a mass of \( 5.75 \, \text{kg} \) and velocity \( 6.00 \, \text{m/s} \) to the right. After they collide, they stick together, and we need to find their combined velocity and the mechanical energy dissipated.

Step 2: Apply Conservation of Momentum

Use conservation of momentum, as the system is closed and isolated. The total initial momentum \( p_{i} \) before the collision is equal to the total final momentum \( p_{f} \) after the collision:\[ m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_f \]Where:\( m_1 = 7.50 \, \text{kg}, \)\( v_1 = -5.00 \, \text{m/s}, \)\( m_2 = 5.75 \, \text{kg}, \)\( v_2 = 6.00 \, \text{m/s} \)Calculate \( v_f \) to find the final velocity after collision.

Step 3: Calculate the Final Velocity

Using the values from Step 2, plug them into the momentum equation:\[ 7.50(-5.00) + 5.75(6.00) = (7.50 + 5.75) v_f \]\[ -37.5 + 34.5 = 13.25 v_f \]\[ -3.0 = 13.25 v_f \]\[ v_f = \frac{-3.0}{13.25} \approx -0.226 \text{ m/s} \]The negative sign indicates the direction is to the left.

Step 4: Calculate Initial Mechanical Energy

Calculate the initial kinetic energy using the formula:\[ KE_{i} = \frac{1}{2}m_1 v_1^2 + \frac{1}{2}m_2 v_2^2 \]\[ KE_{i} = \frac{1}{2}(7.50)(5.00)^2 + \frac{1}{2}(5.75)(6.00)^2 \]\[ KE_{i} = 93.75 + 103.5 = 197.25 \, \text{J} \]

Step 5: Calculate Final Mechanical Energy

Calculate the final kinetic energy using the combined mass and final velocity:\[ KE_{f} = \frac{1}{2}(7.50 + 5.75)(-0.226)^2 \]\[ KE_{f} = \frac{1}{2}(13.25)(0.051) \approx 0.3379 \, \text{J} \]

Step 6: Determine Mechanical Energy Dissipated

Calculate the energy dissipated by subtracting the final kinetic energy from the initial kinetic energy:\[ \Delta KE = KE_{i} - KE_{f} \]\[ \Delta KE = 197.25 - 0.3379 \approx 196.91 \, \text{J} \]So, the energy dissipated is approximately \( 196.91 \, \text{J} \) of mechanical energy.

Key Concepts

Inelastic Collision

An inelastic collision is a type of collision where two objects collide and stick together rather than bouncing off each other. In the context of our exercise with the otters, after slamming into each other on a frictionless surface, they move as a single entity. During inelastic collisions, the law of conservation of momentum still holds.

• Before the collision, each otter has its own momentum, which is calculated by multiplying its mass by its velocity.
• In our example, Otter 1's momentum is headed left at: \( 7.50 \, \text{kg} \times -5.00 \, \text{m/s} = -37.5 \, \text{kg} \cdot \text{m/s} \).
• Otter 2's momentum is headed right at: \( 5.75 \, \text{kg} \times 6.00 \, \text{m/s} = 34.5 \, \text{kg} \cdot \text{m/s} \).

After the collision, both otters hold fast to each other, meaning their combined mass moves with a new velocity. This ensures that the total momentum before the collision equals the total momentum afterward.

Kinetic Energy Dissipation

When the otters collide and stick together, although their total momentum is conserved, kinetic energy is not. Some of this energy converts into other forms such as heat or sound due to deformation upon impact, leading to kinetic energy dissipation.

• We calculate the initial kinetic energy from their separate velocities and masses: for Otter 1: \( K_{1} = \frac{1}{2} \times 7.50 \, \text{kg} \times (5.00 \, \text{m/s})^2 = 93.75 \, \text{J} \).
• Otter 2 contributes: \( K_{2} = \frac{1}{2} \times 5.75 \, \text{kg} \times (6.00 \, \text{m/s})^2 = 103.5 \, \text{J} \).

Total initial kinetic energy is thus \( 197.25 \, \text{J} \). After their collision, the combined kinetic energy is reduced to merely \( 0.3379 \, \text{J} \) due to inelastic behavior. Therefore, a large amount of kinetic energy, precisely \( 196.91 \, \text{J} \), has dissipated.

Frictionless Surface

A frictionless surface is an idealized concept where no friction force acts on the sliding objects. In our exercise with the otters, the muddy surface is assumed frictionless, facilitating a pure demonstration of linear momentum conservation.

• This means that no external horizontal forces (like friction) interfere with the motion of the otters.
• As such, the only forces at play are their mutual gravitational pull and the force they exert on each other during the collision.

On a real-world surface with friction, some momentum would be lost to frictional forces as heat, causing even less kinetic energy available for the motion post-collision. However, with the ideal frictionless surface, the primary focus is purely on the inelastic collision and kinetic energy changes.