An object that is 0.5 m above the ground has the same amount of potential energy as a spring that is stretched 1 m. Each distance is then doubled.

How will the potential energies of the object and the spring compare after the distances are doubled?

A. The gravitational potential energy of the object will be two times greater than the elastic potential energy of the spring.
B. The elastic potential energy of the spring will be four times greater than the gravitational potential energy of the object.
C. The elastic potential energy of the spring will be two times greater than the gravitational potential energy of the object.
D. The potential energies will remain equal to one another.



Answer :

Let's compare the potential energies of both the object and the spring after doubling the distances. Here's a detailed, step-by-step explanation.

### Step 1: Initial Potential Energies

1. Gravitational Potential Energy (GPE)
- Formula: \( GPE = m \cdot g \cdot h \)
- Given:
- \( m = 1 \, \text{kg} \) (mass of the object)
- \( g = 9.8 \, \text{m/s}^2 \) (acceleration due to gravity)
- \( h = 0.5 \, \text{m} \) (initial height above the ground)
- Calculation:
[tex]\[ GPE_{\text{initial}} = 1 \cdot 9.8 \cdot 0.5 = 4.9 \, \text{Joules} \][/tex]

2. Elastic Potential Energy (EPE)
- Formula: \( EPE = \frac{1}{2} k x^2 \)
- Given:
- \( k = 1 \, \text{N/m} \) (spring constant)
- \( x = 0.5 \, \text{m} \) (initial stretch of the spring)
- Calculation:
[tex]\[ EPE_{\text{initial}} = \frac{1}{2} \cdot 1 \cdot (0.5)^2 = 0.125 \, \text{Joules} \][/tex]

At the initial state, we are told that the potential energies are equal.

### Step 2: Doubling the Distances

1. New Gravitational Potential Energy (GPE)
- New height: \( h_{\text{new}} = 2 \times 0.5 = 1 \, \text{m} \)
- Calculation:
[tex]\[ GPE_{\text{new}} = 1 \cdot 9.8 \cdot 1 = 9.8 \, \text{Joules} \][/tex]

2. New Elastic Potential Energy (EPE)
- New stretch: \( x_{\text{new}} = 2 \times 0.5 = 1 \, \text{m} \)
- Calculation:
[tex]\[ EPE_{\text{new}} = \frac{1}{2} \cdot 1 \cdot (1)^2 = 0.5 \, \text{Joules} \][/tex]

### Step 3: Compare the Ratios of the new Potential Energies

1. Ratio of the new GPE to the initial GPE:
[tex]\[ \frac{GPE_{\text{new}}}{GPE_{\text{initial}}} = \frac{9.8}{4.9} = 2 \][/tex]

2. Ratio of the new EPE to the initial EPE:
[tex]\[ \frac{EPE_{\text{new}}}{EPE_{\text{initial}}} = \frac{0.5}{0.125} = 4 \][/tex]

### Step 4: Analysis of Potential Energy Comparison

After doubling the distances, the gravitational potential energy of the object becomes twice its original value, while the elastic potential energy of the spring becomes four times its original value.

Thus, the elastic potential energy of the spring will be four times greater than the gravitational potential energy of the object.

So, the correct answer is:
- The elastic potential energy of the spring will be four times greater than the gravitational potential energy of the object.