Problem 29 - College Physics

Question

A 0.20 -kg object with a horizontal speed of \(10 \mathrm{~m} / \mathrm{s}\) hits a wall and bounces directly back with only half the original speed. (a) What percentage of the object's initial kinetic energy is lost: \((1) 25 \%,(2) 50 \%,\) or (3) \(75 \% ?\) (b) How much kinetic energy is lost in the ball's collision with the wall?

Step-by-Step Solution

Answer

(a) 75%, (b) 7.5 J

Step 1: Calculate Initial Kinetic Energy

To find initial kinetic energy, use the formula \( KE = \frac{1}{2}mv^2 \). Here, \( m = 0.20 \text{ kg} \) and \( v = 10 \text{ m/s} \). Substitute the values: \[ KE_{initial} = \frac{1}{2} \times 0.20 \times (10)^2 = 10 \text{ J}. \]

Step 2: Calculate Final Kinetic Energy

The final velocity is half of the initial velocity. Therefore, \( v = 5 \text{ m/s} \). Use the kinetic energy formula again: \[ KE_{final} = \frac{1}{2} \times 0.20 \times (5)^2 = 2.5 \text{ J}. \]

Step 3: Determine Kinetic Energy Lost

Subtract the final kinetic energy from the initial kinetic energy: \[ KE_{lost} = KE_{initial} - KE_{final} = 10 - 2.5 = 7.5 \text{ J}. \]

Step 4: Calculate Percentage of Kinetic Energy Lost

To find the percentage of kinetic energy lost, use the formula \( \frac{KE_{lost}}{KE_{initial}} \times 100\% \). Plug in the values: \[ \frac{7.5}{10} \times 100\% = 75\%. \]

Key Concepts

Physics Problem Solving

Physics can sometimes seem challenging, but breaking problems down into smaller steps makes solutions more manageable. When approaching a physics problem, it's essential to:

• Understand the problem: Carefully read the exercise to know what is being asked.
• Identify what is given and what needs to be found: Look for values and information provided in the question, and figure out the unknowns.
• Choose the right formulas: Based on the type of problem, select appropriate physics formulas or principles relevant to the solution.
• Solve step by step: Work through the problem methodically, showing each calculation step clearly.

Take the case of the object hitting the wall. The problem involves kinetic energy and speed, so we first use the kinetic energy formula to determine initial and final energy values. Each calculation should be checked along the way to avoid errors. This systematic approach helps in understanding physics concepts more deeply and promotes analytical thinking.

Energy Conservation

Energy conservation is a fundamental concept in physics, stating that energy cannot be created or destroyed. Instead, it only changes from one form to another. In the context of kinetic energy, during a collision, energy can be transformed:

• Some of the object's kinetic energy may transform into heat or sound energy, especially in an inelastic collision where the objects don't bounce perfectly back.
• In perfectly elastic collisions, total kinetic energy is conserved between objects, but this is rare in real-world scenarios.

In our exercise, the object loses a part of its kinetic energy after bouncing back from the wall, indicating an inelastic collision. By calculating the initial and final kinetic energies, we see that energy is not fully conserved in its original form, as a portion is lost, likely to other forms such as sound or heat.

Momentum and Collisions

Momentum, the product of an object's mass and velocity, is a critical concept often coupled with energy conservation in collision problems. In physics, momentum is represented by:

• The formula: \( p = mv \)
• Momentum conservation: In isolated systems with no external forces, total momentum before and after a collision remains constant.

However, during collisions like the object hitting the wall, although the direction of the object's velocity changes, momentum conservation requires analyzing changes in both speed and direction to understand the effects fully.In our exercise, the ball's momentum changes as it reverses direction with reduced speed, indicating that external forces play a role (i.e., the wall). This affects both momentum and energy, aligning with the observed loss in kinetic energy. It's a perfect example of how momentum conservation and kinetic energy changes are interrelated in collision scenarios.