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What is Momentum Help

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By — McGraw-Hill Professional
Updated on Oct 25, 2011

Momentum

Momentum is an expression of “heft in motion,” the product of an object’s mass and its velocity. Momentum, like velocity, is a vector quantity, and its magnitude is expressed in kilogram meters per second (kg · m/s).

Momentum as a Vector

Suppose the speed of an object (in meters per second) is v, and the mass of the object (in kilograms) is m . Then the magnitude of the momentum, p , is their product:

p = mv

This is not the whole story. To fully describe momentum, the direction, as well as the magnitude, must be defined. That means we must consider the velocity of the mass in terms of its speed and direction. (A 2-kg brick flying through your east window is not the same as a 2-kg brick flying through your north window.) If we let v represent the velocity vector and p represent the momentum vector, then we can say this:

p = m v

Impulse

The momentum of a moving object can change in three different ways:

• A change in the mass of the object.
• A change in the speed of the object.
• A change in the direction of the object’s motion.

Let’s consider the second and third of these possibilities together; then this constitutes a change in the velocity.

Fig. 15-10. Three ways in which an impulse can cause a space ship to accelerate. At A, getting the ship to move straight ahead at higher speed. At B and C, getting the ship to turn.

Let’s put our everyday-world time clock into “future mode.” Imagine a massive space ship, coasting along a straight-line path in interstellar space. Consider a point of view, or reference frame, such that the velocity of the ship can be expressed as a vector pointing in a certain direction. A force F can be applied to the ship by firing a rocket engine. Imagine that there are several engines on the ship, one intended for driving it forward at increased speed, and others capable of changing the vessel’s direction. If any engine is fired for t seconds with a force vector of F newtons (as shown by the three examples in Fig. 15-10), then the product F t is called the impulse . Impulse is a vector, symbolized by the uppercase boldface letter I, and its magnitude is expressed in kilogram meters per second (kg · m/s). Here’s the formula:

I = F t

Impulse on an object always produces a change in the object’s velocity. That’s a good thing; it is the purpose of the rocket engines in our space ship! Recall the above formula concerning mass m, force F, and acceleration a:

F = m a

Substitute m a for F in the equation for impulse. Then we get this:

I = ( m a )t

Now remember that acceleration is a change in velocity per unit time. Suppose the velocity of the space ship is v 1 before the rocket is fired, and v 2 afterwards. Then, assuming the rocket engine produces a constant force while it is fired:

a = (v 2 - v 1 )/ t

We can substitute in the previous equation to get

I = m ((v 2v 1 )/ t ) t = m v 2m v 1

This means the impulse is equal to the change in momentum.

We have just derived an important law of real-world motion. Impulse is expressed in kilogram-meters per second (kg · m/s), just as is momentum. You might think of impulse as momentum in another form. When an object is subjected to an impulse, the object’s momentum vector p changes. The vector p can grow larger or smaller in magnitude, or it can change direction, or both of these things can happen.

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