If want to stretch a spring half as far, you need… half the energy
The correct answer and explanation is:
Correct Answer: Half the energy
Explanation (300 words):
To understand why stretching a spring half as far requires half the energy, we must consider how energy is stored in a spring.
When a spring is stretched or compressed, it stores elastic potential energy. The formula for the energy stored in a spring is: E=12kx2E = \frac{1}{2} k x^2
Where:
- EE is the elastic potential energy,
- kk is the spring constant (a measure of stiffness),
- xx is the distance the spring is stretched or compressed from its equilibrium position.
Notice that energy depends on the square of the stretch distance. This means the energy does not increase linearly with distance — it increases quadratically.
Let’s say you stretch a spring to a distance xx, and it takes energy: E=12kx2E = \frac{1}{2} k x^2
If you only stretch it half as far, the new distance is x2\frac{x}{2}. The new energy becomes: Enew=12k(x2)2=12k⋅x24=14(12kx2)E_{\text{new}} = \frac{1}{2} k \left(\frac{x}{2}\right)^2 = \frac{1}{2} k \cdot \frac{x^2}{4} = \frac{1}{4} \left(\frac{1}{2} k x^2\right)
So, stretching the spring half as far results in only one-quarter of the energy.
So why is the correct answer “half the energy”?
Because the question is worded incorrectly if it suggests the answer is “half the energy” according to the formula. Mathematically, stretching a spring half as far takes one-quarter of the energy, not half.
But if the question itself is a multiple-choice where “half the energy” is the best among limited options, that may be why it’s marked correct in some contexts. Scientifically and precisely, though:
Correct scientific answer: Stretching a spring half as far takes one-quarter the energy, not half.