Momentum Help

Introduction

Classical mechanics describes the behavior of objects in motion. Any moving mass has momentum and energy . When two objects collide, the momentum and energy contained in each object changes. We will study the basic newtonian physics.

Momentum

Momentum is the product of an object’s mass and its velocity. The standard unit of mass is the kilogram (kg), and the standard unit of speed is the meter per second (m/s). Momentum magnitude is expressed in kilogram-meters per second (kg · m/s). If the mass of an object moving at a certain speed increases by a factor of 5, then the momentum increases by a factor of 5, assuming that the speed remains constant. If the speed increases by a factor of 5 but the mass remains constant, then again, the momentum increases by a factor of 5.

Momentum As A Vector

Suppose that the speed of an object (in meters per second) is v , and that 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. This means that 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

p = mv

Impulse

The momentum of a moving object can change in any of 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.

Imagine a mass, such as a 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 nonzero vector pointing in a certain direction. A force F may be applied to this vessel by firing a rocket engine. Imagine that there are several engines on this space ship, one intended for driving the vessel forward at increased speed and others capable of changing the vessel’s direction. If any engine is fired for t seconds with force vector of F newtons (as shown by the three examples in Fig. 8-1), then the product F t is called the impulse . Impulse is a vector, symbolized by the uppercase boldface letter I , and is expressed in kilogram meters per second (kg · m/s):

I = F t

Impulse produces a change in velocity. This is clear enough; this is the purpose of rocket engines in a space ship! Recall the 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

Fig. 8-1 . Three different ways in which an impulse can cause an object (in this case, a space ship) to accelerate.

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

a = ( v ₂ − v ₁ )/ t

We can substitute in the preceding equation to get

I = m [( v ₂ − v ₁ )/ t ] t = m v ₂ − m v ₁

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

We have just derived an important law of newtonian physics. Reduced to base units in SI, 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, it can change direction, or both these things can happen.