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Name: Deborah
Status: educator
Age: N/A
Location: NC
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If a particle that is moving has the same momentum and the same kinetic energy as another particle, must their masses and velocities be equal? I do not understand the properties of two particles that have the same momentum, but different kinetic energies.

Hi Deborah,

Momentum is mass times velocity and kinetic energy is one-half mass times velocity squared. If I have a ball with mass M traveling at speed 2V, and a ball of mass 2M traveling at speed V, they both have the same momentum, 2MV. Kinetic energy, however, is a different story. The first ball has kinetic energy of one-half M times the quantity 2V squared or one-half M x 4 V-squared or 2MV-squared. The second ball has kinetic energy one-half 2M x V-squared or MV-squared. The first ball has twice the kinetic energy of the first, even though they have the same momentum.

This explains why when two billiard balls collide with a moving ball striking a stationary ball head-on, the moving ball stops and the stationary ball starts moving at the first ball's velocity. In this case, both momentum and kinetic energy were conserved. If the first ball struck the second and they both took off at half the velocity of the original ball, momentum would be still be conserved but kinetic energy would not. Hope this helps.

Robert Froehlich

Momentum is m*v, and kinetic energy is m*v*v/2, so if momenta and energies are the same, we have:
1) m1*v1 = m2*v2
2) m1*v1*v1 = m2*v2*v2
using (1) in (2) yields
m1*v1*v1 = (m1*v1)*v2 -> v1 = v2
using this in (1) shows that m1 = m2 So, yes, if a particle has the same momentum and the same kinetic energy as another particle, their masses and velocities must be equal

But having the same momentum does not by itself imply having the same energy. A heavy particle moving slowly can have the same momentum as a light particle moving swiftly.

Tim Mooney
Advanced Photon Source, Argonne National Lab.


First, energy and momentum are very different properties.

An object takes time to stop moving. The momentum of an object can be describes as the amount of force required to stop the object in one second. When a constant force is applied to an object, it is the force multiplied by the time over which it is applied that yields the change of momentum of the particle. Also, momentum is a vector: it has a direction to it. Momentum points in the direction of an object's velocity.

An object continues to travel while its velocity drops to zero. The kinetic energy of an object can be described as the amount of force required to stop the object over a distance of one meter. When a constant force is applied to an object, it is the force multiplied by distance traveled along the SAME AXIS as the force that determines the change of kinetic energy. Kinetic energy is a scalar: it has no direction.

An object changing direction but neither speeding up nor slowing down is an example of changing momentum but not changing kinetic energy. If the object does speed up or slow down, both momentum and kinetic energy will change. For example, consider throwing a rock upward at a certain speed. If you double the rock's initial speed, the rock will require twice the time and four times the distance to reach zero speed. Thus, the rock with the doubled speed has twice the momentum and four times the kinetic energy.

Dr. Ken Mellendorf
Physics Instructor
Illinois Central College

Dear Deborah,

Yes as a little algebra should convince you, if two particles have the same momentum and the same kinetic energy, their masses and speeds must be the same.

Just take the equation m1 v1^2 = m2 v2^2 (equal kinetic energies) and divide by m1 v1 = m2 v2 (equal momenta). You get v1 = v2. Put that in m1 v1 = m2 v2 and you get m1 = m2.

To understand two particles that have the same momentum but different kinetic energies, write the kinetic energy as p^2/(2m). So if two particles have the same momentum (p), their kinetic energies are proportional to the inverse of their masses.

Best, Dick Plano, Professor of Physics emeritus, Rutgers University

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