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Derive the formula of Kinetic Energy.?
Most Upvoted Answer
Derive the formula of Kinetic Energy.?
Kinetic energy is a simple concept with a simple equation that is simple to derive. Let's do it twice.
Derivation using algebra alone (and assuming acceleration is constant). Start from the work-energy theorem, then add in Newton's second law of motion.
ΔK = W = FΔs = maΔs
Take the the appropriate equation from kinematics and rearrange it a bit.
| v 2 = v0 2 + 2aΔs | |
| aΔs = | v 2 − v0 2 |
| 2 |
Combine the two expressions.
| ΔK = m | ⎛ ⎝ | v 2 − v0 2 | ⎞ ⎠ |
| 2 |
And now something a bit unusual. Expand.
| ΔK = | 1 | mv 2 − | 1 | mv 0 2 |
| 2 | 2 |
If kinetic energy is the energy of motion then, naturally, the kinetic energy of an object at rest should be zero. Therefore, we don't need the second term and an object's
kinetic energy
is just…K = �mv
2
Derivation using calculus (but now we don't need to assume anything about the acceleration). Again, start from the work-energy theorem and add in Newton's second law of motion (the calculus version).
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Rearrange the differential terms to get the integral and the function into agreement.
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The integral of which is quite simple to evaluate over the limits initial speed (
v
) to final speed (v
).0
| ΔK = | 1 | mv 2 − | 1 | mv 0 2 |
| 2 | 2 |
Naturally, the kinetic energy of an object at rest should be zero. Thus an object's
kinetic energy
is defined mathematically by the following equation…K = �mv
2
Community Answer
Derive the formula of Kinetic Energy.?
Kinetic Energy Formula:
The formula for kinetic energy is given by:
KE = 1/2 * m * v^2
Where:
- KE represents kinetic energy
- m represents the mass of the object
- v represents the velocity of the object
Explanation:
Kinetic energy is the energy possessed by an object due to its motion. It is a scalar quantity and depends on both the mass and velocity of the object. The formula for kinetic energy can be derived using basic principles of physics.
Derivation:
1. Consider an object of mass 'm' moving with a velocity 'v'.
2. The work done on the object to accelerate it from rest to its current velocity is equal to the change in its kinetic energy.
3. The work done, W, is defined as the force applied, F, multiplied by the displacement, s, in the direction of the force.
4. According to Newton's second law of motion, the force applied to an object is given by F = ma, where a is the acceleration of the object.
5. The displacement can be expressed as s = vt, where t is the time taken to reach the current velocity.
6. Substituting F = ma and s = vt into the work formula, we get W = mad.
7. The equation for acceleration can be rearranged as a = v/t.
8. Substituting the value of acceleration into the equation for work, we have W = m(v/t)d.
9. The distance covered, d, can be expressed as d = vt, where v is the average velocity.
10. Substituting the value of distance into the equation for work, we get W = mv^2.
11. The work done on the object is equal to the change in its kinetic energy, so W = ΔKE.
12. Therefore, ΔKE = mv^2.
13. As the object is initially at rest, the change in kinetic energy is equal to the final kinetic energy, KE.
14. Hence, KE = mv^2.
15. To calculate the kinetic energy, we divide the formula by 2 to account for the fact that the object starts from rest. Therefore, KE = 1/2 * m * v^2.
Conclusion:
The formula for kinetic energy, KE = 1/2 * m * v^2, represents the energy possessed by an object due to its motion. It is derived from the principles of work and energy, considering the change in kinetic energy as the work done on the object. The formula involves the mass and velocity of the object, with the square of the velocity having a significant impact on the object's kinetic energy.
The formula for kinetic energy is given by:
KE = 1/2 * m * v^2
Where:
- KE represents kinetic energy
- m represents the mass of the object
- v represents the velocity of the object
Explanation:
Kinetic energy is the energy possessed by an object due to its motion. It is a scalar quantity and depends on both the mass and velocity of the object. The formula for kinetic energy can be derived using basic principles of physics.
Derivation:
1. Consider an object of mass 'm' moving with a velocity 'v'.
2. The work done on the object to accelerate it from rest to its current velocity is equal to the change in its kinetic energy.
3. The work done, W, is defined as the force applied, F, multiplied by the displacement, s, in the direction of the force.
4. According to Newton's second law of motion, the force applied to an object is given by F = ma, where a is the acceleration of the object.
5. The displacement can be expressed as s = vt, where t is the time taken to reach the current velocity.
6. Substituting F = ma and s = vt into the work formula, we get W = mad.
7. The equation for acceleration can be rearranged as a = v/t.
8. Substituting the value of acceleration into the equation for work, we have W = m(v/t)d.
9. The distance covered, d, can be expressed as d = vt, where v is the average velocity.
10. Substituting the value of distance into the equation for work, we get W = mv^2.
11. The work done on the object is equal to the change in its kinetic energy, so W = ΔKE.
12. Therefore, ΔKE = mv^2.
13. As the object is initially at rest, the change in kinetic energy is equal to the final kinetic energy, KE.
14. Hence, KE = mv^2.
15. To calculate the kinetic energy, we divide the formula by 2 to account for the fact that the object starts from rest. Therefore, KE = 1/2 * m * v^2.
Conclusion:
The formula for kinetic energy, KE = 1/2 * m * v^2, represents the energy possessed by an object due to its motion. It is derived from the principles of work and energy, considering the change in kinetic energy as the work done on the object. The formula involves the mass and velocity of the object, with the square of the velocity having a significant impact on the object's kinetic energy.
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