QUESTION #462

# How is force related to momentum?

Asked by: Melissa Thomas

### Answer

Momentum measures the 'motion content' of an object, and is based on the product of an object's mass and velocity. Momentum doubles, for example, when velocity doubles. Similarly, if two objects are moving with the same velocity, one with twice the mass of the other also has twice the momentum.

Force, on the other hand, is the push or pull that is applied to an object to CHANGE its momentum. Newton's second law of motion defines force as the product of mass times ACCELERATION (vs. velocity). Since acceleration is the change in velocity divided by time, you can connect the two concepts with the following relationship:

force = mass x (velocity / time) = (mass x velocity) / time = momentum / time

Multiplying both sides of this equation by time:

force x time = momentum

To answer your original question, then, the difference between force and momentum is time. Knowing the amount of force and the length of time that force is applied to an object will tell you the resulting change in its momentum.
Answered by: Paul Walorski, B.A., Part-time Physics Instructor

### Answer

They are related by the fact that force is the rate at which momentum changes with respect to time (F = dp/dt). Note that if p = mv and m is constant, then F = dp/dt = m*dv/dt = ma. On the other hand, you can also say that the change in momentum is equal to the force multiplied by the time in which it was applied (or the integral of force with respect to time, if the force is not constant over the time period).

Interestingly enough, this, along with Newton's Third law, gives us conservation of momentum. Newton's Third law says that for a force exerted by object 1 on object 2, object 2 exerts a force on object 1 that is equal in magnitude and opposite in direction to the force object 1 exerts. Or, more succinctly, F[1->2] = -F[2->1]. Now the total change in momentum for any interaction is the integral of F[1->2] over time plus the integral of F[2->1] over time, which equals the integral of F[1->2] minus the integral of F[1->2], which equals zero - always!

A similar argument for conservation of energy can be made using the fact that energy is the integral of force with respect to position.
Answered by: Gregory Ogin, Physics Undergraduate Student, UST, St. Paul, MN

### Answer

Newton's 2nd Law tells us that force = mass x acceleration ( F = ma ). Since acceleration is just how velocity changes over time, we can write this as

F = m * v/t

Multiply both sides by time to arrive at

F t = m v

Since mv is momentum, we can see that the momentum conferred to an object by a force equals the force times the time the force is applied. Thus if a 15 Newton force to the right is applied to an initially stationary object for 3 seconds, it will have a momentum of 45 kg m/s to the right.

Most students who ask this question are usually trying to figure out the reverse situation, however. If an object hits me with a certain amount of momentum, how much force does it hit me with? Note that due to Newton's 3rd Law, this can be calculated the same way. If a thrown egg hits your hand with a momentum of 5 kg m/s, the force it applies to your hand depends on the time it takes for your hand to absorb the momentum. If you hold your hand very stiffly (and try to make the egg stop in a very short period of time) the ball exerts a high force on your hand, e.g. 100 N for 1/20th of a second. However as anyone who has ever played in an egg toss knows, if you let your hand 'give' and extend the amount of time it takes to absorb the momentum, the egg exerts a smaller force on your hand, e.g. 10 N for 1/2 a second.
Answered by: Rob Landolfi, Science Teacher, Washington, DC