Question

The potential energy of a particle is given by $U(x)=20+(x-2)^{2}$ where U is in joules and x in meters. The minimum potential energy and the position where it occurs are:}

• A. 20 J at $x=2$ m
• B. 2 J at $x=20$ m
• C. 22 J at $x=2$ m
• D. 0 J at $x=2$ m

Hint:

This is in the vertex form of a parabola $y = k + (x-h)^2$. The vertex (minimum) is $(h, k)$.

Answer 1

The Correct Option is A

Step 1: Concept
Minimum value of a quadratic function occurs when the squared term is zero.

Step 2: Meaning
In $U(x)=20+(x-2)^{2}$, the term $(x-2)^{2}$ is always $\ge 0$.

Step 3: Analysis
The smallest possible value for $(x-2)^2$ is 0, which happens at $x=2$.

Step 4: Conclusion
At $x=2$, $U = 20 + 0 = 20$ J.
Final Answer: (A)