Problem 18 - University Physics with Modern Physics

Question

A slingshot will shoot a \(10-8\) pebble 220 \(\mathrm{m}\) straight up. (a) How much potential energy is stored in the slingshot's rubber band? (b) With the same potential energy stored in the rubber band, how high can the slingshot shoot a \(25-g\) pebble? (c) What physical effects did you ignore in solving this problem?

Step-by-Step Solution

Answer

(a) 21.56 J, (b) approximately 88 m, (c) ignored air resistance and rubber band efficiency.

Step 1: Identify the Given Information

First, identify the given information to solve the problem: The initial mass of the pebble is 10 g, which is equivalent to 0.01 kg, and it is shot up to a height of 220 m. For part (b), the mass of the second pebble is 25 g, or 0.025 kg.

Step 2: Calculate Potential Energy Stored

Potential energy when the 10 g pebble reaches its maximum height can be calculated using the gravitational potential energy formula: \[ PE = mgh \]Where \( m = 0.01 \) kg, \( g = 9.8 \) m/s², and \( h = 220 \) m.Substitute the values:\[ PE = 0.01 \times 9.8 \times 220 = 21.56 \text{ J} \]

Step 3: Find Maximum Height for New Pebble

Use the calculated potential energy of 21.56 J to find the maximum height the slingshot can shoot a 25 g pebble. Formula is the same,\[ h = \frac{PE}{mg} \]Substituting the values:\[ h = \frac{21.56}{0.025 \times 9.8} \approx 88 \text{ m} \]

Step 4: Consider Ignored Effects

Recognize any effects ignored during this calculation, which include air resistance, deformation of the rubber band (efficiency factor), wind resistance, and rotational effects on the pebble.

Key Concepts

Gravitational Potential Energy

Gravitational potential energy is a type of energy that an object possesses because of its position in a gravitational field. It depends on three primary factors:

• Mass of the object ( m)
• Gravitational acceleration ( g), which is approximately 9.8 m/s² on Earth
• Height above a reference point ( h)

The formula to calculate gravitational potential energy ( PE) is:\[PE = mgh\]This equation shows that the energy is directly proportional to the mass and height of the object. In our case, for a pebble of 10 g reaching a height of 220 m, the potential energy stored is 21.56 Joules. This amount of energy was accumulated in the rubber band of the slingshot, allowing it to launch the pebble to the specified height.

Energy Conservation

The principle of energy conservation states that energy cannot be created or destroyed, only transformed from one form to another. In this context, the potential energy stored in the slingshot's rubber band is entirely converted into gravitational potential energy when the pebble reaches its maximum height.

The initial potential energy in the slingshot remains the same when shooting different pebbles. For a new 25 g pebble shot with the same energy of 21.56 J, the maximum height it can reach is determined by:\[h = \frac{PE}{mg}\]Using this formula, we find that the pebble can reach approximately 88 meters. This showcases how the principle of energy conservation is central in determining the outcome of energy transformation in such scenarios.

Physics Problem Solving

Approaching physics problems with a structured plan can simplify complex tasks. Here’s a useful method to tackle similar problems:

• Identify what you are solving for and list all the known values.
• Use the appropriate physics principles or formulas related to the problem.
• Substitute known values into the formula and solve for the unknown.
• Review the result to ensure it makes sense in the context of the problem.

In this exercise, we carefully identified the masses and used the gravitational potential energy formula. By doing this step-by-step, it facilitated finding the solutions efficiently and accurately. Adequate understanding of the core physics concepts is crucial in solving such exercises effectively.

Physics Concepts Ignored in Calculations

In this physics problem, certain real-world effects were overlooked to simplify calculations. Ignoring these factors is common in introductory physics exercises to focus on key concepts:

• Air Resistance: The frictional force air exerts against a moving object, which can reduce the maximum height reached by the pebble.
• Deformation of the Rubber Band: The slingshot material's efficiency might reduce the actual potential energy transfer.
• Wind Resistance: Any wind present can alter the path and height of the pebble.
• Rotational Effects: If the pebble spins during flight, it can affect stability and height.

Understanding these ignored factors is critical as they can significantly affect outcomes in more precise and advanced scenarios. Recognizing them helps in comprehending the limitations of theoretical models.