Physics Problem: Ball Energy Conversion

This problem asks us to find the final speed of a ball just before it hits the ground. We know its mass and its potential energy when it was held at a certain height before being released to fall freely.

Key Physics Concepts: Energy Conservation

To solve this, we use the principle of Conservation of Mechanical Energy. This principle states that if we ignore forces like air resistance, the total mechanical energy of an object remains constant. Mechanical energy is the sum of potential energy (energy due to position) and kinetic energy (energy due to motion).

• Potential Energy (PE): The energy stored in the ball due to its height. It's calculated as PE = mgh.
• Kinetic Energy (KE): The energy the ball gains as it falls and speeds up. It's calculated as \(KE = \frac{1}{2}mv^2\).
• Energy Transformation: As the ball falls, its potential energy decreases, and this lost potential energy is converted into kinetic energy, causing the ball's speed to increase.

Calculating Ball Speed: Step-by-Step

1. List Known Values:
   • Mass (m): \(320 \text{ g}\). Convert this to kilograms (the standard unit for physics calculations): \(m = \frac{320}{1000} \text{ kg} = 0.320 \text{ kg}\).
   • Initial Potential Energy (\(PE_{initial}\)): \(625 \text{ J}\). This is the energy the ball has due to its height before falling.
   • Initial Kinetic Energy (\(KE_{initial}\)): Since the ball is released freely, it starts from rest. Therefore, its initial speed is \(0 \text{ m/s}\), and \(KE_{initial} = 0 \text{ J}\).
   • Final Potential Energy (\(PE_{final}\)): When the ball hits the ground, its height is essentially 0. Thus, its potential energy at that point is \(PE_{final} = 0 \text{ J}\).
2. Apply the Conservation of Energy Principle:

   According to the conservation of energy, the total mechanical energy at the start must equal the total mechanical energy at the end.

   \(PE_{initial} + KE_{initial} = PE_{final} + KE_{final}\)

3. Determine Final Kinetic Energy:

   Substitute the known values into the conservation equation:

   \(625 \text{ J} + 0 \text{ J} = 0 \text{ J} + KE_{final}\)

   This simplifies to:

   \(KE_{final} = 625 \text{ J}\)

   This result shows that all the initial potential energy has been converted into kinetic energy just before the ball hits the ground.

4. Calculate the Final Speed (v):

   Now, we use the formula for kinetic energy to find the speed:

   \(KE_{final} = \frac{1}{2}mv^2\)

   Substitute the values for \(KE_{final}\) and m:

   \(625 \text{ J} = \frac{1}{2} \times (0.320 \text{ kg}) \times v^2\)

   Simplify the equation:

   \(625 = 0.160 \times v^2\)

   Rearrange the equation to solve for \(v^2\):

   \(v^2 = \frac{625}{0.160}\)

   \(v^2 = 3906.25 \text{ (m/s)}^2\)

   Finally, take the square root to find the speed v:

   \(v = \sqrt{3906.25} \text{ m/s}\)

   \(v = 62.5 \text{ m/s}\)

Final Ball Speed Calculation

The calculation confirms that the speed of the ball when it hits the ground is \(62.5 \text{ m/s}\).