You can find kinetic energy using momentum through the formula KE = p²/2m, where KE is kinetic energy, p is momentum, and m is mass. This formula is derived by substituting the velocity from the momentum formula into the standard kinetic energy formula.
Understanding the Derivation
Here's a breakdown of how the formula KE = p²/2m is derived:
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Momentum Formula: The momentum (p) of an object is given by the equation:
- p = mv
where m is the mass and v is the velocity of the object.
- p = mv
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Solving for Velocity: We can rearrange this formula to solve for velocity:
- v = p / m
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Kinetic Energy Formula: The standard formula for kinetic energy (KE) is:
- KE = (1/2)mv²
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Substitution: Now, substitute the velocity expression (v = p/m) into the kinetic energy formula:
- KE = (1/2)m(p/m)²
- KE = (1/2)m(p²/m²)
- KE = (1/2)(p²/m)
- KE = p²/2m
How to Use the Formula
To calculate kinetic energy using momentum, simply:
- Find the momentum of the object.
- Find the mass of the object.
- Square the momentum.
- Divide the squared momentum by twice the mass.
Example
Let's say an object has:
- Momentum (p) of 10 kg·m/s
- Mass (m) of 2 kg
To calculate the kinetic energy (KE):
- KE = p²/2m
- KE = (10 kg·m/s)² / (2 * 2 kg)
- KE = 100 kg²·m²/s² / 4 kg
- KE = 25 Joules
Therefore, the kinetic energy of the object is 25 Joules.
Key Takeaways
- The formula KE = p²/2m is essential for finding kinetic energy using momentum.
- This formula combines the concepts of momentum and kinetic energy.
- Understanding the derivation helps in comprehending the relationship between these two concepts.
| Concept | Formula |
|---|---|
| Momentum (p) | p = mv |
| Kinetic Energy (KE) | KE = (1/2)mv² |
| KE with Momentum | KE = p²/2m |
