How is energy conserved in physics problems?

Conservation of Energy: Definition

In physics, the law of conservation of energy states that energy cannot be created or destroyed; it can only be transformed from one form to another or transferred between objects. The total energy of an isolated system remains constant over time.

Mathematically, for a closed system:

$$ E_{\text{total, initial}} = E_{\text{total, final}} $$

where $E_{\text{total}}$ includes all forms of energy (kinetic, potential, thermal, etc.).

Worked Example: Block Sliding Down a Frictionless Ramp

Problem:
A block of mass $m$ starts from rest at height $h$ on a frictionless ramp. Find its speed $v$ at the bottom.

Step 1: Identify Energies

• At the top: Only gravitational potential energy ($U$)
• At the bottom: Only kinetic energy ($K$)

Step 2: Write the Energy Conservation Equation

$$ U_{\text{top}} + K_{\text{top}} = U_{\text{bottom}} + K_{\text{bottom}} $$

At the top: $U_{\text{top}} = mgh$, $K_{\text{top}} = 0$
At the bottom: $U_{\text{bottom}} = 0$, $K_{\text{bottom}} = \frac{1}{2}mv^2$

So,

$$ mgh + 0 = 0 + \frac{1}{2}mv^2 $$

Step 3: Solve for $v$

$$ mgh = \frac{1}{2}mv^2 $$

Divide both sides by $m$ (assuming $m \neq 0$):

$$ gh = \frac{1}{2}v^2 $$

Multiply both sides by $2$:

$$ 2gh = v^2 $$

Take the square root:

$$ v = \sqrt{2gh} $$

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Takeaways

• The total energy in a closed system remains constant; energy changes form but is not lost.
• Conservation of energy allows us to solve for unknown quantities without knowing all the forces involved.
• In the absence of non-conservative forces (like friction), mechanical energy (kinetic + potential) is conserved.