Question

\(\textbf{Combining Conservation Laws}\). A 5.00-kg chunk of ice is sliding at 12.0 m/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 \((\textbf{Fig. P8.73})\). Since the valley is icy, there is no friction. After the collision, how high above the valley floor will the combined chunks go?

Short answer

The chunks will rise approximately 1.835 meters above the valley floor.

Step-by-step solution

Understand the Problem

We have two chunks of ice, each weighing 5.00 kg. The first chunk is moving at 12.0 m/s, and the second chunk is at rest. They collide and stick together, and we need to find how high the combined mass rises post-collision given there's no friction.

Conservation of Momentum

Initial momentum before the collision is \( m_1 \times v_1 + m_2 \times v_2 \), where \( m_1 = 5.00 \) kg, \( v_1 = 12.0 \) m/s, and \( v_2 = 0 \) m/s for the stationary chunk. Total initial momentum is \( 5.00 \times 12.0 + 5.00 \times 0 = 60.0 \, \text{kg m/s} \). According to the principle of conservation of momentum, the momentum after collision is also \( 60.0 \, \text{kg m/s} \).

Calculate Final Velocity Post-Collision

Using conservation of momentum, \( m_1v_1 + m_2v_2 = (m_1 + m_2)v_f \), where \( v_f \) is the final velocity. Solve for \( v_f \): \[ 60.0 = (5.00 + 5.00)v_f \] \[ v_f = \frac{60.0}{10.0} = 6.0 \text{ m/s} \].

Conservation of Energy

Now apply conservation of energy to find the height. The kinetic energy post-collision \( KE = \frac{1}{2}(m_1 + m_2)v_f^2 \) is converted to gravitational potential energy \( PE = (m_1 + m_2)gh \) at the peak height: \[ \frac{1}{2}(10.0)(6.0)^2 = 10.0 \times 9.81 \times h \]. Solve for \( h \).

Solve for Height

Substitute and solve for height:\[ \frac{1}{2} \times 10.0 \times 36.0 = 10.0 \times 9.81 \times h \]\[ 180.0 = 98.1h \]\[ h = \frac{180.0}{98.1} \approx 1.835 \text{ meters} \].

Key concepts

Conservation of Momentum

In physics, the conservation of momentum principle is incredibly useful when analyzing collisions. Momentum, which is the product of an object's mass and velocity, is conserved in isolated systems where no external forces act upon the interacting bodies. This means that the total momentum before and after a collision remains the same. In the case of the ice chunks, the initial momentum was calculated as the product of 5.00 kg and 12.0 m/s for the moving chunk, resulting in 60.0 kg m/s. Given that the second chunk was at rest, it contributed nothing to the initial momentum. After the collision, the combined mass moved with a velocity that we calculated using the conservation of momentum, demonstrating how the momentum of the system remained unchanged.

Conservation of Energy

The concept of energy conservation is pivotal in understanding how energy transforms from one form to another without loss in closed systems. In our ice chunk scenario, after the collision, the kinetic energy of the moving mass was transformed into gravitational potential energy as it ascended from the valley floor. The principle of conservation of energy allowed us to equate the initial kinetic energy with the gravitational potential energy at the apex of the trajectory. Prior to reaching the maximum height, the kinetic energy calculated as 108 J (joules) was then used to find out how high the combined mass would rise, illustrating how energy is conserved and traded between kinetic and potential forms without dissipation, given the frictionless environment.

Collision Physics

Collision physics encompasses the study of interactions between objects that exert forces on each other for a relatively short time. There are different types of collisions: elastic, where both momentum and kinetic energy are conserved, and inelastic, where merely momentum is conserved. In the problem with the ice chunks, we have an inelastic collision because the chunks stick together after impact. The kinetic energy isn't conserved due to heat, sound, and slight deformations, but momentum is. By utilizing the conservation of momentum, we could ascertain the combined velocity post-collision providing insights on how the chunks would behave once they stuck together.

Kinetic Energy

Kinetic energy is the energy an object possesses due to its motion, calculated as \( \frac{1}{2}mv^2 \). This formula reveals how both mass and velocity contribute to kinetic energy. In the ice chunk example, right after the collision, the chunks move together at a speed of 6.0 m/s, resulting in kinetic energy that later gets converted to potential energy. Understanding kinetic energy provides clarity on how moving objects work, and why changes in speed or mass lead to changes in the energy stored within the system, allowing predictions about motion post-collision.

Gravitational Potential Energy

Gravitational potential energy is the energy due to an object's position in a gravitational field, primarily dependent on its height above a reference point. The formula, \( PE = mgh \), shows how mass and height above the ground level affect the stored energy. For the chunks of ice, the initial kinetic energy they had post-collision was used to lift them to a maximum height. At this peak, all kinetic energy was converted into gravitational potential energy, allowing for the calculation of height. This energy conversion elucidates how potential energy is stored in height and provides insight into the maximum height an object can reach driven by initial kinetic energy.