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Conservation of Energy Formula Calculator

\[ \frac{1}{2}mv^2 + mgh = \frac{1}{2}mv_{\text{final}}^2 + mgh_{\text{final}} \]

1. What is the Conservation of Energy Formula Calculator?

Definition: This calculator computes the final velocity (\(v_{\text{final}}\)) of an object using the conservation of mechanical energy, given its mass (\(m\)), initial velocity (\(v\)), initial height (\(h\)), and final height (\(h_{\text{final}}\)).

Purpose: It is used in physics to analyze the motion of objects under gravity, such as falling objects or roller coasters, where mechanical energy (kinetic + potential) is conserved.

2. How Does the Calculator Work?

The calculator uses the conservation of mechanical energy:

Formula: \[ \frac{1}{2}mv^2 + mgh = \frac{1}{2}mv_{\text{final}}^2 + mgh_{\text{final}} \] Solving for \(v_{\text{final}}\): \[ v_{\text{final}} = \sqrt{v^2 + 2g(h - h_{\text{final}})} \] where:

  • \(v_{\text{final}}\): Final velocity (m/s, km/h, mph)
  • \(v\): Initial velocity (m/s, km/h, mph)
  • \(m\): Mass (kg, g, mg)
  • \(g\): Gravitational acceleration (m/s², default 9.80665)
  • \(h\): Initial height (m, km, cm)
  • \(h_{\text{final}}\): Final height (m, km, cm)

Unit Conversions:

  • Mass:
    • 1 kg = 1 kg
    • 1 g = 0.001 kg
    • 1 mg = 0.000001 kg
  • Velocity:
    • 1 m/s = 1 m/s
    • 1 km/h = 0.277778 m/s
    • 1 mph = 0.44704 m/s
  • Height:
    • 1 m = 1 m
    • 1 km = 1000 m
    • 1 cm = 0.01 m

Steps:

  • Enter the mass in kg, g, or mg (default 1 kg, step size 0.00001).
  • Enter the initial velocity in m/s, km/h, or mph (default 0 m/s, step size 0.00001).
  • Enter the initial height in m, km, or cm (default 10 m, step size 0.00001).
  • Enter the final height in m, km, or cm (default 0 m, step size 0.00001).
  • Enter the gravitational acceleration in m/s² (default 9.80665, step size 0.00001).
  • Convert inputs to base units (kg, m/s, m).
  • Validate that mass and gravity are positive, and check if the final height is reachable.
  • Calculate final velocity: \(v_{\text{final}} = \sqrt{v^2 + 2g(h - h_{\text{final}})}\).
  • Convert the final velocity to the selected unit.
  • Display the result, using scientific notation if the absolute value is less than 0.001, otherwise rounded to 2 decimal places.

3. Importance of Conservation of Energy Calculation

Calculating using conservation of energy is crucial for:

  • Physics: Analyzing motion in systems where mechanical energy is conserved, such as pendulums or roller coasters.
  • Engineering: Designing systems like amusement rides or vehicles, ensuring safe speeds at different heights.
  • Education: Teaching the principle of energy conservation in mechanics and dynamics.

4. Using the Calculator

Examples:

  • Example 1: Calculate the final velocity for \(m = 1 \, \text{kg}\), \(v = 0 \, \text{m/s}\), \(h = 10 \, \text{m}\), \(h_{\text{final}} = 0 \, \text{m}\), \(g = 9.80665 \, \text{m/s}^2\), in m/s:
    • Enter \(m = 1 \, \text{kg}\), \(v = 0 \, \text{m/s}\), \(h = 10 \, \text{m}\), \(h_{\text{final}} = 0 \, \text{m}\), \(g = 9.80665 \, \text{m/s}^2\).
    • Final velocity: \(v_{\text{final}} = \sqrt{0^2 + 2 \times 9.80665 \times (10 - 0)} = \sqrt{196.133} \approx 14.00 \, \text{m/s}\).
    • Result: \( \text{Final Velocity} = 14.00 \, \text{m/s} \).
  • Example 2: Calculate the final velocity for \(m = 500 \, \text{g}\), \(v = 10 \, \text{m/s}\), \(h = 50 \, \text{cm}\), \(h_{\text{final}} = 20 \, \text{cm}\), \(g = 9.80665 \, \text{m/s}^2\), in km/h:
    • Enter \(m = 500 \, \text{g}\), \(v = 10 \, \text{m/s}\), \(h = 50 \, \text{cm}\), \(h_{\text{final}} = 20 \, \text{cm}\), \(g = 9.80665 \, \text{m/s}^2\).
    • Convert: \(m = 0.5 \, \text{kg}\), \(h = 0.5 \, \text{m}\), \(h_{\text{final}} = 0.2 \, \text{m}\).
    • Final velocity: \(v_{\text{final}} = \sqrt{10^2 + 2 \times 9.80665 \times (0.5 - 0.2)} = \sqrt{100 + 5.88399} \approx 10.29 \, \text{m/s} \approx 37.04 \, \text{km/h}\).
    • Result: \( \text{Final Velocity} = 37.04 \, \text{km/h} \).

5. Frequently Asked Questions (FAQ)

Q: What does conservation of energy mean in this context?
A: Conservation of energy means that the total mechanical energy (kinetic + potential) of an object remains constant if no non-conservative forces (like friction) are acting.

Q: Why might the final velocity calculation fail?
A: If the final height is higher than the initial height and the initial velocity is insufficient, the object cannot reach that height, resulting in an imaginary velocity.

Q: Why is mass not used in the final velocity calculation?
A: Mass cancels out in the conservation of energy equation, so the final velocity depends only on velocity, height difference, and gravity.

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