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 kg, is sliding to the left at 5.00 m/s, while the other, of mass 5.75 kg, is slipping to the right at 6.00 m/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?

Short answer

(a) The final velocity is approximately -0.227 m/s to the left. (b) About 196.91 J of mechanical energy dissipates.

Step-by-step solution

Identify Given Data

We have two otters, one with mass \( m_1 = 7.50 \text{ kg} \) moving left at \( v_1 = -5.00 \text{ m/s} \), and another with mass \( m_2 = 5.75 \text{ kg} \) moving right at \( v_2 = 6.00 \text{ m/s} \). After the collision, they stick together and move with a common velocity \( v_f \).

Apply Conservation of Momentum

The law of conservation of momentum states that the total momentum before the collision equals the total momentum after the collision. We set up the equation as follows: \[ m_1 v_1 + m_2 v_2 = (m_1 + m_2) v_f \]Substituting the known values: \[ (7.50 \cdot -5.00) + (5.75 \cdot 6.00) = (7.50 + 5.75) v_f \]

Solve for Final Velocity

Calculate each term:\[ -37.5 + 34.5 = 13.25 v_f \]\[ -3.0 = 13.25 v_f \]Now, solve for \( v_f \):\[ v_f = \frac{-3.0}{13.25} \approx -0.227 \text{ m/s} \]The velocity is negative, indicating the direction of the velocity is to the left.

Calculate Initial Mechanical Energy

Calculate the initial kinetic energy (KE) for both otters:\[ KE_1 = \frac{1}{2} m_1 v_1^2 = \frac{1}{2} \times 7.50 \times (-5.00)^2 = 93.75 \text{ J} \]\[ KE_2 = \frac{1}{2} m_2 v_2^2 = \frac{1}{2} \times 5.75 \times 6.00^2 = 103.5 \text{ J} \]The total initial mechanical energy is:\[ KE_{initial} = 93.75 + 103.5 = 197.25 \text{ J} \]

Calculate Final Mechanical Energy

The final kinetic energy of the combined mass is given by:\[ KE_{final} = \frac{1}{2} (m_1 + m_2) v_f^2 = \frac{1}{2} \times 13.25 \times (-0.227)^2 \approx 0.34135 \text{ J} \]

Calculate Energy Dissipated

The mechanical energy dissipated due to the collision is the difference between the initial and final kinetic energy:\[ \Delta KE = KE_{initial} - KE_{final} = 197.25 - 0.34135 \approx 196.91 \text{ J} \]

Key concepts

Collision Mechanics

In collision mechanics, especially on a frictionless horizontal surface, understanding the motion and interaction of objects is crucial. When two objects, like our playful otters, collide and stick together, the type of collision is termed a "perfectly inelastic collision." This means that after colliding, their separate movements cease, and they continue moving together as a single entity.

In perfect inelastic collisions, the concept of momentum is key. The momentum of a system is conserved even if kinetic energy is not, which is a famous principle known as the conservation of momentum. Before the collision, each otter has its distinct momentum determined by its velocity and mass. Despite their different directions, the total momentum before impact equals the total momentum after impact.

• Momentum before collision: \( m_1 v_1 + m_2 v_2 \)
• Momentum after collision: \( (m_1 + m_2) v_f \)

What makes the calculation special here is the opposite direction of the initial velocities, rendering one negative. This principle aids not only in physics problems involving collisions but also in real-life applications like vehicle crash analyses and sports mechanics.

Kinetic Energy

Kinetic energy (KE) is the energy an object possesses due to its motion. For any moving object, the formula for kinetic energy is \( KE = \frac{1}{2} mv^2 \). This formula highlights how kinetic energy is dependent both on mass and the square of velocity. Hence, even a small speed increase results in significantly more kinetic energy.

At the start of our otters' journey, each possesses its own kinetic energy. The speeds at which they approach each other determine the initial amounts of energy at play:

• Otter 1 with mass 7.50 kg and velocity -5.00 m/s has a kinetic energy of 93.75 J.
• Otter 2 with mass 5.75 kg and velocity 6.00 m/s has a kinetic energy of 103.5 J.
• Total initial kinetic energy is the sum: 197.25 J.

After they collide and move together, we observe that the kinetic energy of the system is much lower (0.34135 J). This reduction happens because some energy is not just transferred into motion but also dissipated elsewhere, like sound or internal energy of deformation.

Mechanical Energy Dissipation

Mechanical energy dissipation is a crucial aspect in understanding real-world collisions. While momentum is conserved, kinetic energy often isn't, particularly in inelastic collisions. This dissimilarity between momentum and kinetic energy outcomes in such collisions aligns with the idea of mechanical energy being transformed into other forms.

When the fun-loving otters collide and eventually move as one, the energy lost is not vanished, but is instead redirected to other forms such as heat, sound, or internal energy increase among the objects involved in the collision. For this scenario, the initial mechanical energy was 197.25 J and reduced drastically to 0.34135 J after the collision. The energy that disappeared from the kinetic realm equals 196.91 J. This energy loss is critical, demonstrating how not all kinetic energy during inelastic collisions is used effectively in moving the objects forward post-collision.

• This energy loss often leads to heating up of the objects and surrounding air, noise from the collision, and any deformation that may occur.
• Understanding energy dissipation is vital in designing systems like car crumple zones where such energy redirection can save lives by ensuring less kinetic energy is transferred to passengers.