Do we really need “linear momentum”? Can’t we just find a way to study motion with velocity, inertia, force, or energy? Why bring more jargon into what is already a confusing mess of terminologies and equations?
To a beginner, the concept of momentum can be a difficult thing to grasp and sometimes confusing. For example, momentum is a vector quantity, meaning it is a quantity that has magnitude as well as direction in space. This alone brings in a cascade of problems when solving questions because both have to be taken into account. And then comes along the concept of “conservation of momentum”, which is like spilling gasoline on the confusion fire. It doesn’t help that physics teachers love to mess around with these ideas as they lay traps and snares into their student’s exams.
So why is there a need for such a mischievous quantity as momentum in the first place?
It is necessary. I’ll give you an example…
Suppose you want to break your friend’s car window – for reasons that you yourself know. If you are really keen on smashing the glass window, you’d be careful to select a stone of the right size and weight. Not just any stone; you’d pick one that is big enough to cause the required damage but not too big, lest you fail to throw it fast enough.
Simply put, you will be looking to maximize the damage by selecting the best combination of weight and velocity; a stone that is heavy enough to cause damage but one you can throw fast enough.
Therefore, if we want to get a sense of the measure of “impactfulness” that a body in motion would have upon another, taking some combination of weight and velocity of the moving body would be an ideal place to start.
But how can we combine weight and velocity into one quantity that makes sense?
Congratulations! You are now thinking like a scientist
Sir Isaac Newton asked himself this very question some 350 years ago as he was contemplating the motion of bodies. He realized that the motion of a body depends greatly on its mass and velocity. And although velocity is a good measure of how fast a body is moving through space, it doesn’t tell us anything about the “will-power” of the moving body. For example, a truck and a supermarket trolley can both be moving at 5mph, but you would rather be knocked over by a supermarket trolley than the truck!
Velocity is a purely kinematic quantity; it doesn’t tell us anything about how a body would respond to an external influence – such as a stone smashing into a window for example.
To properly study the dynamics of motion, Isaac Newton conceived a new physical quantity.
Linear momentum
Isaac Newton came up with what he called, “quantity of motion” – which he defined to be the product of mass and velocity of a body in motion. By “quantity of motion”, Newton was essentially talking about the will-power or impactfulness of a body in motion as we have discussed.
Why mass and velocity?
Because they are what matters most when quantifying motion in the manner we discussed. Think about it, when a body is in motion what do you want to know?
• How fast it’s moving?
• What direction is it headed?
• Where is it? (i.e. its position in space at a given time)
• How heavy is it?
• What color??
• Is the body happy?? (I dunno)
Velocity answers the first two of those questions. It tells us:
• How fast a body is moving – this is speed, the magnitude of velocity.
• What direction is it headed – velocity as a vector quantity tells us where the body is headed.
What about position? You may ask.
Well, this is not an important question when strictly dealing with motion. One just needs to know how fast/ slow a body in motion changes its position, not its exact position at all times.
How heavy is it – this is where “mass” comes in. And it’s important that we use mass and not weight!
As you can see, it ultimately comes down to mass and velocity when analyzing the motion of an object. The rest of the questions (color, shape, type, etc.) are irrelevant.
What is linear momentum then?
Physicists define the product of mass and the velocity of a moving body as its linear momentum. The term linear is used to indicate the type of motion i.e. motion along a straight line.
In mathematical terms:
p⃗ =mv⃗
Where p= the linear momentum of a body in motion
m= the mass of the body in motion
v = the velocity of the body in motion.
Generally speaking, the product (mass x velocity) tells you about the potency of a body in motion (not just how fast it’s moving).
For example, if you want to know how long it’ll take to drive coast-to-coast, you only need to know how fast the vehicle is moving. However, if you want to assess your options when breaks suddenly fail on the highway, you’ll need to know if you’d be driving a light vehicle or a freight truck.
Linear momentum and direction
Linear momentum is a vector quantity – it has both magnitude and direction. You can think of it like an arrow flying out, it’s always pointing somewhere. The direction of momentum takes the same as the direction of the velocity.
If a car is speeding off in the north direction from your position, then its momentum (mass x velocity) points in the north direction and so forth. Momentum is not complete without indicating direction.
Why is it important that linear momentum has direction?
Let us revisit our original example, the one where you are so pissed off at your friend you want to break her car window! So you are standing over her car at the dawn of day, holding a brick in your hand, how would you throw the brick?
Common sense dictates that you throw it perpendicular to the surface of the car window so that the glass window sits directly along the trajectory of the stone. In this arrangement, the stone inevitably has to go through the glass (shattering it in the process) and into the car. If the stone impacts the glass window at an angle; it may bounce or slide off – laying your efforts to waste (alerting the neighbors in the process!). Direction is just as important as mass and speed when looking to maximize the “impactfulness” that a body in motion will have on another.
Floyd Mayweather’s jab may weigh 60 pounds and can come at you clocking at 25 miles an hour, but if he miscalculates direction and fails to make good contact – it all goes to waste. The potency of a body in motion depends very much on the direction as in mass and speed.
Related: Understanding the parallelogram law of vector addition
Applications of linear momentum in our lives
Amanda* couldn’t believe what was happening, she had just lost her young daughter in a tragic car accident when a truck broadsided her car. She and her baby son had barely escaped with their lives. Yet she now found herself staring at a time in prison for her own role in the accident. Not only did she have to live with the tragedy of losing her daughter, but she was also about to lose whatever she had been left with if found guilty. The prosecution was charging her with operating to endanger vehicular homicide. If convicted, she was sure to do jail time for the crime.
On that fateful day, Amanda had made an illegal U-turn shortly before a truck slammed onto the side of her car – killing her daughter. She knew she wasn’t totally innocent, but she was convinced that if the truck driver had been a little more careful, he could have avoided the collision.
To prove her case, she consulted an accident reconstruction expert to help visualize the scene of the accident and determine the chain of events that led to the collision. An accident reconstruction expert assesses everything that may have contributed to the crash such as speed, low visibility, weather conditions, etc. The expert’s testimony gives an objective account based on science and facts, which can be very useful in deciding a case where liability is contested or multiple parties may be at fault.
One of the most important tools of an accident reconstruction expert is momentum. During an accident, two or more bodies in motion collide into one another, with each body impressing impact on the other consistent with its momentum. An expert can use this knowledge to recreate a snap-shot, frame by frame, of what exactly happened during the collision.
And this is exactly what happened in Amanda’s case, using the physics of dynamics of motion, the expert worked out the speed at which the truck driver slammed onto Amanda’s car, which he found to be much higher than the speed-limit on that highway. He went on to deduce that, if the truck driver had been going at the correct speed, Amanda would have cleared his lane in time or he would have had enough reaction-time to avoid a collision.
The grieving mother was found not guilty of motor vehicle homicide as well as not guilty of operating to endanger. She was discharged from her arrest and free to go.
[Disclaimer] This is based on a true story taken from a case study of Eden Rafferty Law Firm. Amanda is not the real name of the victim in this story. You can read the original article here: Grieving mother found not guilty of vehicular homicide in an emotional trial.
One cannot overstate the enormous application of momentum (and other physics concepts) in our daily lives. This is just one example of how momentum was used to potentially prevent a wrongful conviction. This article won’t be enough to cover all the useful applications of linear momentum in other fields of our lives.
Related: What is angular momentum in layman’s terms, with real-life applications
