Below, we show that this equation is equivalent to the more familiar version of Newton's second law based on mass and acceleration. We state
that version of Newton’s law, and then use the definition to restate the law as you see it here.
Step Reason
1. Newton’s second law; definition of acceleration
2. p = mv definition of momentum
3. divide equation 2 by ǻt
4. substitute equation 3 into equation 1
7.3 - Impulse
Impulse:Change in momentum.
In the prior section, we stated that net force equals change in momentum per unit time.
We can rearrange this equation and state that the change in momentum equals the
product of the average force and the elapsed time, which is shown in Equation 1.
The change in momentum is called the impulse of the force, and is represented by J.
Impulse is a vector, has the same units as momentum, and points in the same direction
as the change in momentum and as the force. The relationship shown is called the
impulse-momentum theorem.
In this section, we focus on the case when a force is applied for a brief interval of time,
and is stated or approximated as an average force. This is a common and important
way of applying the concept of impulse.
A net force is required to accelerate an object, changing its velocity and its momentum.
The greater the net force, or the longer the interval of time it is applied, the more the
object's momentum changes, which is the same as saying the impulse increases.
Engineers apply this concept to the systems they design. For instance, a cannon barrel
is long so that the cannonball is exposed to the force of the explosive charge longer,
which causes the cannonball to experience a greater impulse, and a greater change in
momentum.
Even though a longer barrel allows the force to be applied for a longer time interval, it is
still brief. Measuring a rapidly changing force over such an interval may be difficult, so
the force is often modeled as an average force. For example, when a baseball player
swings a bat and hits the ball, the duration of the collision can be as short as 1/1000th of
a second and the force averages in the thousands of newtons.
The brief but large force that the bat exerts on the ball is called an impulsive force. When analyzing a collision like this, we ignore other forces
(like gravity) that are acting upon the ball because their effect is minimal during this brief period of time.
In the illustration for Equation 1, you see a force that varies with time (the curve) and the average of that force (the straight dashed line). The
area under the curve and the area of the rectangle both equal the impulse, since both equal the product of force and time.
The nature of impulse explains why coaches teach athletes like long jumpers, cyclists, skiers and martial artists to relax when they land or fall,
and why padded mats and sand pits are used. In Example 1 on the right, we calculate the (one-dimensional) impulse experienced by a long
jumper on landing in the sand pit, from her change in momentum.
Impulse
Average force times elapsed time
Change in momentum