**Introduction**

In this lesson, we will be studying objects in motion from the point of view of the velocity and mass they carry, This concept we call *momentum*. One more time we will find a connection between motion and interaction: in this case, a more complex concept—*impulse*, Based on this knowledge, we will discuss an important law of conservation — momentum conservation — and apply it to elastic and perfect inelastic collisions.

**Linear Momentum**

From driving, biking, rollerblading, or any other motion-related activities, you might recall that a heavier object and a lighter object collide with other objects in different ways. If I recall my first lesson in bowling: You have to let go of the bowling ball while you are moving instead of while you are simply standing, because the ball will have more *momentum*. Do you think I had a good teacher? We define momentum as the product of the mass and its velocity at one time, and we say that momentum measures inertia for an object in motion.

p = *m* · v

Momentum

This quantity is a vector and its direction is the same direction as the velocity. Also, the unit for momentum is kg · m/s.

**Example**

Two different objects have the same momentum, but one object is ten times larger in mass than the other. How do the two velocities compare? Consider onedimensional motion.

**Solution**

The assumption suggested by the problem refers to the expression of the momentum where both momentum and velocity are vectors. If we consider a 1-D motion, then the significance of the vector sign disappears.

*m*_{1} = 10 · *m*_{2}

*p*_{1} = *p*_{2}

*v*_{1}/*v*_{2} = ?

*p*_{1} = *m*_{1} · *v*_{1}

*p*_{2} = *m*_{2} · *v*_{2}

So, the second object having a smaller mass moves faster at the same momentum.

**Impulse**

In order to move an object, a net force applied to the object is necessary. The *magnitude* of the force and the *interval of time* that the force acts on the object are important in determining the effect. Figure 6.1 shows a measurement of the time dependence of force in two separate collisions.

On the left, the effect on the object is much larger than on the right, because for the same interval of time, the average force applied is larger.

In order to fully characterize this process, a quantity called *impulse* is introduced and the *impulse of a force*, *J*, is proportional to the product of the average force acting on the object and the interval of time of contact between the two:

**J** = **F** · **Δ** *t*

The direction of the impulse is the same as the direction of the average force, and the unit for impulse is N · seconds. Note that because the force is time dependent, we cannot consider any random point on the curve, but must use an average for the entire time interval instead.

**Example**

Consider a soccer player hitting a ball with an average force of about 0.80 kN. Find the impulse if the contact between the ball and the player's foot extends to about 6.0 ms.

**Solution**

Consider first the data given in the problem and the units of the data. Then a simple replacement of the quantities in the expression of impulse will yield the answer.

F* _{average}* = 0.80 kN = 800 N

**Δ** *t* = 6.0 ms = 6.0 · 10 ^{– 3} s

*J* = ?

According to the definition, the impulse is the product of the average force and the time interval:

*J* = F · **Δ** *t*

*J* = 800N · 6.0 · 10^{– 3} s

*J* = 4.8 Ns

The direction of the impulse is the same as the direction of the force.

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