Chapter 8: Momentum, Change in Momentum and Impulse

Equations Introduced and Used for this Topic (all equations can be written and solved as both scalar and vector and all equations are generally solved as vectors):

Impulse (J)  = Ft = ∆pChange in Momentum (∆p) = m∆v

Momentum (p) = mv

∆p = p_f – p_i∆v = v_f – v_i

Where

J   is the Impulse or the product of Force x Time acting on a system, measured in either Newton-Seconds (N_s) or Kilograms Metres per Second (kg m/s)

p   Momentum is the product of Mass times Velocity and is measured in either Newton-Seconds (N_s) or Kilograms Metres per Second (kg m/s)

∆ p   is the Change in Momentum (p_f – p_i or m(v_f -v_i), measured in either Newton-Seconds (N_s) or Kilograms Metres per Second (kg m/s)

m   is Mass, commonly measured in Kilograms (Kg), Grams (g) or Tonnes (t)

F   is Force, measured in Newtons (N)

t  is Time, commonly measured in Seconds (s) or Hours (h)

∆v is the change in an object’s speed generally taken as the difference between the final and initial speed, measured in metres per second (m/s) or kilometres per hour (km/h)

v   is Average Speed, commonly measured in Metres/Second (m/s) or Kilometres/Hour (km/h)

Impulse and Momentum ^{4, 5} are two very important concepts that can be easily derived from Newton’s Second Law and the equation defining acceleration, specifically using:

F = m a                     &                        [latex]a\dfrac{=\Delta{v}}{t}[/latex]

When you substitute for a in Newton’s Second Law, you are left with:

[latex]F = m\dfrac{\Delta{v}}{t}[/latex]

Which is more commonly written as:

F t = m ∆v

Written in this form we now have both the equations for impulse (F t) and change in momentum (m ∆v). Both impulse and change in momentum are important physics concepts and have important implications in multiple areas such as analyzing collisions, predicting crash injuries and even helping to analyze rockets and space travel.

If you check out the units used in either F t or in m ∆v, you get two new unit combinations:

N_s and kg m/s  (or kg ms^-1)

These units are equivalent and can be used for either measure, with the most common unit being N_s because it is quicker to write down.  They are interchangeable and you should expect them to be switched around without explanation.

In this chapter we are also adding these two measures of Impulse (J = F t) and Change in Momentum (∆p = m ∆v) to Chapters 1 – 6 (our study of mechanics) and as such will be using all equations and variables that have been used up to this point.

8.1 Linear Momentum ⁶

Momentum (p) = mv

Momentum is used in common discussions and generally refers to the concept that once something is in motion or is moving then it becomes difficult to change. You can hear this referred to the momentum of a game, where one team starts to dominate the other team, or in politics, where one politician or party is gaining strength and can challenge the leader or is simply increasing its lead over the others.

In physics, momentum is a very specific term that is used in both vector and scalar situations.

It is the product of the mass of the object times the speed or the velocity of the object.  This product gives us a very specific measure of this object and allows us to calculate multiple analyses.

We all have a logical understanding of momentum. All you need to ask yourself is:

Would you rather be struck by a fly traveling at 15 km/h or bicyclist traveling at 15 km/h?

Your answer is telling you that momentum is the most likely the reason for your choice.

These first few examples are restricted to working with a constant or an average momentum.

8.2 Change in Momentum

Impulse (J)  = Ft = ∆pChange in Momentum (∆p) = m∆v

Momentum (p) = mv

∆p = p_f – pi∆v = v_f – v_i

NASA photo of the space shuttle Endeavour leaving Earth taken from a low orbit observation jet aircraft that was most likely traveling at 10.7 km or lower. Space travel is currently only possible as a result of the ejection of high temperature gases, which can be looked at as impulse causing the spacecraft to move opposite to the direction of the ejected gas.

Change in Momentum,  like change in displacement or velocity, can be found by calculating the difference between the final momentum and the initial momentum. The next few examples cover questions that explore this concept.

8.3 Impulse and Change in Momentum ^{7, 8, 9}

In the introduction to chapter 7, we derived both the Impulse and Change in Momentum equations, combining Newton’s Second Law and Acceleration, resulting in:

F = m a    &   a =  [latex]\dfrac{\Delta{v}}{t}[/latex] which yielded F t = m ∆v

It is the combined equation of  F t = m ∆v  that allows one to measure the change in momentum that results from an impulse or the impulse needed to cause a change in momentum.  This is a different approach to finding the change in momentum by using the difference of the final and initial momentum, ∆p = p_f – p_i or by using the change in speed/velocity ∆p = m(v_f – v_i).  This combined equation also uses the versatility of Newton’s Second Law to expand tools that are useful in explaining other natural phenomena.

Impulse and the resulting change in momentum manifest themselves in many different forms in our physical world. In contact sports, most athletes now wear protective headgear, such as helmets, and in many parts of the world the law requires that people wear such helmets when riding scooters, bicycles and motorcycles.  Headgear like helmets will extend the time of a force acting on a person’s head if they encounter a blow. Since the change in the momentum remains the same, if the time of the blow increases, the force of the blow is decreased. This lowers the chance of severe brain injuries.

Cushioning and reducing the impact of blows are not restricted to contact sports but include cushioned surfaces and mats for individuals involved in sports such as gymnastics, cheerleading and wrestling or other martial arts. Injuries increase in severity if surfaces where the individual impacts the surface are solid and have no cushioning surface that can absorb and reduce the acting forces. Automobiles built prior to 1959 did not have seat belts or air bags that are currently used as safety features to reduce driver and passenger injuries. As a result, passenger mortality began to dramatically decrease post 1959 due to increased focus on ways one could reduce the impact of injuries on vehicle occupants.

While the solution to the above example was elementary, the next few questions use the relationship between impulse and change in momentum to arrive at a solution. Using the relationship that

Impulse = Change on Momentum

or

F t = m ∆v

we now have two equations that can be used to find either the Impulse or Change in Momentum. They can be combined to find the Force, Change in Velocity (or Speed), Time or Mass, all of which depend on what is being asked and if enough data is available.

Working with this, – 16 m/s change in velocity requires one to subtract 16 m/s from the initial velocity of 22.2 m/s south. Since 22.2 m/s south is greater than the 16 m/s that is removed due to braking, the car is still in motion.

Had the change in velocity been greater than the initial 22.2 m/s south, then the car would have been brought to a full stop.

This final example uses the relationship between Impulse and Change in Momentum to add other possible methods for solving the four kinematic equations covered in Chapter 3.  As before, the variables are defined in exactly the same way as in the previous chapters, and the chart introduced in Chapter 3 remains useful in solving these types of problems.

You now have more tools at your disposal to solve these kinematic problems.

Consider the following example: