Problem 79 - University Physics with Modern Physics

Question

Combining Conservation Laws. A 5.00 -kg chunk of ice is sliding at 12.0 \(\mathrm{m} / \mathrm{s}\) on the floor of an ice-covered valley when it collides with and sticks to another 5.00 -kg chunk of ice that is initially at rest. Fig. \(P 8.79\) ). Since the valley is icy, there is no friction. After the collision, how high above the valley floor will the combined chunks go?

Step-by-Step Solution

Answer

The combined chunks will rise to a height of 1.83 meters.

Step 1: Analyze the Situation Using Conservation of Momentum

Before the collision, we have only one ice chunk moving with a velocity of 12.0 m/s and the other chunk at rest. Using conservation of momentum, the total momentum before collision is equal to the total momentum after collision.The momentum before the collision is given by:\[ p_i = m_1 v_1 = 5.00 \, \text{kg} \times 12.0 \, \text{m/s} = 60.0 \, \text{kg m/s} \]Both chunks stick together, so the combined mass after the collision is:\[ m_f = m_1 + m_2 = 5.00 \, \text{kg} + 5.00 \, \text{kg} = 10.00 \, \text{kg} \]The initial velocity of the system after the collision \( v_f \) can be found using:\[ p_f = m_f v_f \Rightarrow 60.0 \, \text{kg m/s} = 10.00 \, \text{kg} \times v_f \]Solving for \( v_f \):\[ v_f = \frac{60.0}{10.00} = 6.0 \, \text{m/s} \]

Step 2: Use Conservation of Energy to Find Maximum Height

The kinetic energy of the system after the collision will be converted to gravitational potential energy at the highest point when the chunks stop moving upward. The expression for kinetic energy \( KE \) after the collision is:\[ KE = \frac{1}{2} m_f v_f^2 = \frac{1}{2} \times 10.00 \, \text{kg} \times (6.0 \, \text{m/s})^2 = 180 \, \text{J} \]At the maximum height, all kinetic energy is converted into potential energy \( PE \):\[ PE = m_f g h \]Set \( KE = PE \):\[ 180 \, \text{J} = 10.00 \, \text{kg} \times 9.81 \, \text{m/s}^2 \times h \]Solving for \( h \):\[ h = \frac{180}{10.00 \times 9.81} = 1.83 \, \text{m} \]

Key Concepts

Conservation of Energy

Conservation of energy is a fundamental principle in physics stating that energy cannot be created or destroyed; it can only change forms. In the context of the ice chunks colliding and moving upwards, we see a beautiful interplay between kinetic and potential energy. As the combined ice chunks rise, their kinetic energy, which is the energy due to motion, is converted into potential energy, the energy stored due to their position. Ultimately, at the highest point of their rise, all the kinetic energy is transformed into potential energy.

• This energy transformation helps us calculate the maximum height the chunks reach. When they're at their peak height, they momentarily stop before gravity pulls them back down, indicating that all the kinetic energy has become potential energy.
• In this exercise, conservation of energy helps in finding out how high the ice chunks climb after their collision. By equating the kinetic energy after collision with the potential energy at the highest point, we find the height they reach.

Collisions

Collisions are events where two or more bodies exert forces on each other for a short time, leading to a change in their motion. In physics, especially in problems like the ice chunks collision, we apply the principle of conservation of momentum—a close cousin of the conservation of energy. During a perfectly inelastic collision like this one, where the chunks stick together, the total momentum of the system remains unchanged, even though kinetic energy might not.

• Before the collision, one ice chunk was moving while the other was stationary. This led to a scenario where just before impact, the total momentum was the product of the mass and velocity of the moving chunk.
• After the collision, the two chunks act as a single object with a combined mass, moving at a common velocity. The momentum is conserved, but the energy takes different forms.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, described by the equation: \[ KE = \frac{1}{2} mv^2 \]where \( m \) is mass and \( v \) is velocity. After the ice chunks collide and stick together, their motion post-collision can be analyzed using kinetic energy. The equation helps us calculate how much of the system's energy is in motion and is critical in understanding energy conversions.

• Once the collision has occurred, the two ice chunks move with a new velocity, and calculating the kinetic energy at this stage allows us to determine how much energy is in play as this kinetic energy eventually becomes potential energy when the chunks move upwards.
• This conversion of kinetic energy to potential energy allows us to predict the maximum height they achieve.

Potential Energy

Potential energy is the stored energy of an object resulting from its position or configuration. For the ice chunks, this is primarily gravitational potential energy as they move vertically in the absence of resistive forces due to friction.

• In mathematical terms, it is expressed as \[ PE = mgh \]where \( m \) is mass, \( g \) is the acceleration due to gravity, and \( h \) is height.
• Initially, all the kinetic energy from the moving chunks is transferred to potential energy when they reach their highest point. Knowing this not only helps us calculate their potential energy, but it also shows how they will fall back down as potential energy is reconverted to kinetic energy.

Understanding how potential energy functions in situations like these illustrates the seamless conversion between different types of energy, governed by the conservation laws.