Conservation of Energy Calculator
This calculator applies the conservation of mechanical energy to compare two states of a moving object. Enter the mass, height and velocity at state 1 and the height at state 2 to find the velocity at state 2 and verify that total energy is conserved.
Calculate Total Mechanical Energy
Conservation of Energy Formula
Where:
- \(m\) — mass of the object (kg)
- \(v_1\), \(v_2\) — velocity of the object at state 1 and state 2 (m/s)
- \(h_1\), \(h_2\) — height of the object at state 1 and state 2 (m)
- \(g\) — gravitational acceleration (9.81 m/s\(^2\) on Earth)
- \(KE = \tfrac{1}{2}mv^2\) — kinetic energy (J)
- \(PE = mgh\) — gravitational potential energy (J)
This equation states that the total mechanical energy \(E = KE + PE\) stays constant when an object moves without friction or air resistance: whatever potential energy is lost during a fall reappears as kinetic energy, and vice versa.
Conservation of Energy – Calculation Example
Example: a 5 kg object is released from rest at a height of 10 m and falls freely to the ground (\(h_2 = 0\)). Find its speed just before impact.
\[ m = 5\ \text{kg},\quad h_1 = 10\ \text{m},\quad v_1 = 0\ \text{m/s},\quad h_2 = 0\ \text{m},\quad g = 9.81\ \text{m/s}^2 \]
Energy at state 1 (all potential, since \(v_1 = 0\)):
\[ KE_1 = \tfrac{1}{2}\cdot 5 \cdot 0^2 = 0\ \text{J},\qquad PE_1 = 5 \cdot 9.81 \cdot 10 = 490.5\ \text{J},\qquad E_1 = 490.5\ \text{J} \]
Solving the conservation equation for \(v_2\):
\[ v_2 = \sqrt{\frac{2\,(E_1 - mgh_2)}{m}} = \sqrt{\frac{2 \cdot (490.5 - 0)}{5}} = \sqrt{196.2} \approx 14.007\ \text{m/s} \]
Result: the total energy is 490.5 J in both states, and the object hits the ground at \(v_2 \approx 14.007\) m/s (about 50.4 km/h).
How to Use the Calculator
- Enter the Mass (m) of the object and choose the unit (kg or g).
- Under State 1, fill in Height 1 (h₁) and Velocity 1 (v₁) with their units (m or cm for height, m/s or km/h for velocity). State 1 is the starting point of the motion.
- Under State 2, enter Height 2 (h₂) — the point you want to analyze. You can also change the unit of Velocity 2 (v₂); its value is computed by the calculator, and the unit you pick only sets how the answer is displayed.
- Check Gravitational Acceleration (g). The default 9.81 m/s\(^2\) applies to Earth; change it for the Moon (1.62) or another body if needed.
- Press Calculate. The calculator shows the total energy at state 1, the velocity at state 2, and the total energy at state 2 — the two energy values should match, confirming that energy is conserved.
What Is Conservation of Energy?
The law of conservation of energy is one of the most fundamental principles in physics: energy cannot be created or destroyed, only transformed from one form to another. In mechanics, the two forms that matter most are kinetic energy — the energy of motion — and gravitational potential energy, which depends on an object's height above a reference level. Their sum, the mechanical energy, remains constant as long as no energy leaves the system.
A falling object illustrates the idea perfectly. At the top of its path it moves slowly and sits high above the ground, so most of its energy is potential. As it falls, height decreases and speed increases: potential energy converts into kinetic energy at exactly the same rate. At the moment of impact, all the original potential energy has become kinetic energy. Roll a ball up a ramp and the process runs in reverse.
Because the total is fixed, you can predict the speed at any height without knowing the details of the path. A ball thrown straight up, a puck sliding down a frictionless curve, and a pendulum swinging all obey the same equation — the speed depends only on the energy balance, not on the shape of the trajectory. That is why conservation of energy is such a powerful problem-solving tool: it replaces a detailed force analysis with a simple bookkeeping of energy at two states.
When interpreting the result, remember that both energies are measured relative to the chosen height reference. The calculator treats \(h = 0\) as the level where potential energy is zero, and the total energy values are only meaningful relative to that same reference. What must match between the two states is the total, not the individual kinetic or potential parts.
Assumptions and Limitations
- No friction or air resistance. The calculator assumes a perfectly conservative system. With drag or friction present, the real velocity at state 2 will be lower than the calculated value; for falling through air, see the Terminal Velocity Calculator.
- Rigid body, no rotation. Only translational kinetic energy is included. If the object rolls or spins, part of the energy goes into rotation — the Rotational Kinetic Energy Calculator covers that case.
- Constant gravitational acceleration. The value of \(g\) is treated as the same at both heights, which is an excellent approximation near the Earth's surface.
- Height reference is arbitrary. You may set \(h = 0\) anywhere; only differences in height affect the result. Keep the same reference for both states.
- If the numbers you enter imply more energy at state 2 than at state 1 (for example, a higher position and a higher speed with no energy input), the term under the square root becomes negative and no physical solution exists — check your inputs.
Practical Tips
- For a pure free-fall problem, set \(v_1 = 0\) and \(h_2 = 0\); the answer then equals the classic free-fall speed \(v = \sqrt{2gh}\). The Free Fall Calculator handles the full time-dependent motion.
- To find how high a thrown object will climb, set \(v_2 = 0\) conceptually by leaving state 2 with a velocity of zero — enter the launch speed at state 1 and solve for the height where all energy is potential.
- The same principle governs pendulum swings and roller-coaster hills; for related motion problems, try the Projectile Motion Calculator and the Simple Pendulum Calculator.
- Real machines always lose some energy to friction and heat. To quantify how much useful energy survives, use the Mechanical Efficiency Calculator.
Frequently Asked Questions
How do I calculate conservation of energy by hand?
Write the energy balance \(\tfrac{1}{2}mv_1^2 + mgh_1 = \tfrac{1}{2}mv_2^2 + mgh_2\), plug in the known values, and solve for the unknown. For velocity: \(v_2 = \sqrt{v_1^2 + 2g(h_1 - h_2)}\). Compute each term in joules first (kinetic and potential separately) to check your arithmetic — the two totals must match, otherwise an input is wrong.
Why does the mass cancel out for free fall?
If an object starts at rest, mass appears on both sides of the energy equation and divides out, giving \(v = \sqrt{2g\Delta h}\). A feather and a hammer released from the same height in a vacuum hit the ground at the same speed. Mass still matters in this calculator for the absolute energy values in joules — a heavier object stores more energy at the same height and speed.
What does a negative number under the square root mean?
If your inputs imply state 2 needs more energy than state 1 (higher position and higher speed) with no energy input, the equation has no physical solution — the motion is impossible. Check the heights: the classic mistake is mixing units or switching the two states. Also confirm \(h_1\) and \(h_2\) use the same zero reference level.
Does the height reference (h = 0) matter?
No — only the height difference affects the velocity. You can set zero at the ground, the tabletop, or even below the object; the computed \(v_2\) is identical. The absolute energy values in joules do change with the reference, which is why two people can report different total energies for the same problem and both be correct.
What value of g should I use for the Moon, Mars, or other planets?
Earth: \(g \approx 9.81\ \text{m/s}^2\) (9.78 at the equator, 9.83 at the poles). The Moon: 1.62. Mars: 3.71. Jupiter: 24.79. Simply replace the default value in the gravitational acceleration field. On the Moon, the same drop from 10 m gives only about 5.7 m/s instead of 14 m/s.
When does conservation of mechanical energy not apply?
Whenever energy leaks out of mechanical form: friction, air resistance, inelastic collisions, or an engine doing work. Then \(KE_2 + PE_2\) will be less than \(KE_1 + PE_1\), and the real speed at state 2 is lower than the calculator's ideal value. For everyday falls shorter than a few hundred meters, air resistance causes only a small error for compact objects.
Why is the speed at a given height independent of the path?
Because gravitational force is conservative: the energy balance only compares the start and end heights, not the trajectory between them. A ball sliding down a frictionless ramp, a pendulum bob, and a freely falling object all reach the same speed at the same drop height. This is exactly what makes energy methods faster than analyzing forces along a curved path.