Chapter M1: Momentum
The Physics of Collisions 🚗💥
Introduction
1. Introduction
Alright, let's dive into one of the most fundamental concepts in Mechanics: Momentum! Ever wondered what makes a bowling ball so much more effective than a tennis ball at knocking over pins, even if you threw them at the same speed? That's momentum in action. In this chapter, we're going to break it all down. First, we'll get to grips with Linear Momentum itself: what it is and how to calculate it (). Then, we'll explore the super-important Principle of Conservation of Momentum, which is the key to every Paper 4 question about the direct impact of two bodies moving along one straight line, including bodies that coalesce. Finally, we'll see how to find the kinetic energy lost in a collision and how to handle questions where a collision sits in the middle of a longer story. Let's get started!
2. Definition and Vector Nature of Linear Momentum
Alright, let's dive into Linear Momentum. Think of it as the 'quantity of motion' an object has, or its 'oomph'. It's not just about how fast something is going, but also how heavy it is. A bowling ball rolling slowly can have the same momentum as a tennis ball moving super fast. The formula is beautifully simple: , where is momentum, is mass, and is velocity.
Now, here’s the crucial part for your A-Level exams: momentum is a vector. This means it has both magnitude (its size) and direction. In Paper 4 everything moves along one straight line, so "direction" is just a sign: choose one direction along the line as positive and write it down, and momentum takes the same sign as the velocity.
Now, here’s the crucial part for your A-Level exams: momentum is a vector. This means it has both magnitude (its size) and direction. In Paper 4 everything moves along one straight line, so "direction" is just a sign: choose one direction along the line as positive and write it down, and momentum takes the same sign as the velocity.
So, if we define 'right' as the positive direction, an object moving left has a negative velocity and therefore a negative momentum. Two bodies with momenta and have a total momentum of : adding the sizes to get would describe nothing real. The standard unit for momentum is the kilogram-metre per second, written as kg m s⁻¹. You will also see the newton second (N s), which is the same unit because N kg m s⁻².
When a body's velocity changes from to , its change in momentum is , with both velocities put in with their signs. If the body keeps going the same way, the size of the change is mass times the difference of the speeds; if it reverses (a rebound), the size is mass times the sum of the speeds.
When a body's velocity changes from to , its change in momentum is , with both velocities put in with their signs. If the body keeps going the same way, the size of the change is mass times the difference of the speeds; if it reverses (a rebound), the size is mass times the sum of the speeds.
Worked example
Worked Example: Momentum of Each Body and of the Pair
Let's Crunch Some Momentum Numbers 🔢
A car of mass kg moves along a straight road at m s⁻¹ to the east. A truck of mass kg moves along the same road at m s⁻¹ to the west. Taking east as positive, find the momentum of each vehicle and the total momentum of the two vehicles.
- 1First, turn every direction word into a sign. East is positive, so the car's velocity is positive and the truck's is negative. Our go-to formula is .
- 2Now find each momentum, keeping the sign.
- 3The total momentum is the signed sum, not the sum of the sizes.
- 4Finally, state the answer in words, with a direction. The truck is heavier and its momentum is larger in size, so the total should point the truck's way. It does. Don't lose easy marks by forgetting units or direction!Car: kg m s⁻¹ east. Truck: kg m s⁻¹ west. Total: kg m s⁻¹ west.
Answer
Car: kg m s⁻¹ east. Truck: kg m s⁻¹ west. Total: kg m s⁻¹ west.
3. The Principle of Conservation of Linear Momentum
Alright, let's get into one of the most fundamental laws in all of mechanics: the Conservation of Momentum. When two particles collide, the force each one puts on the other is equal and opposite (Newton's third law), and both forces act for the same short time, so whatever momentum one particle gains, the other loses. The core idea is this: in a collision, the total momentum before is equal to the total momentum after.
Mathematically, for two particles A and B in a direct impact (meaning they collide head-on along a straight line), we express this as:
Here, is mass, is the initial velocity, and is the final velocity. The most common mistake people make is forgetting that velocity has a direction. Your first step in any problem should be to define a positive direction (e.g., to the right). Any velocity going in the opposite direction must be given a negative sign. If an after-velocity is unknown, write it as and let the sign of your answer tell you the direction. Get this right, and you're halfway to the solution.
Mathematically, for two particles A and B in a direct impact (meaning they collide head-on along a straight line), we express this as:
Here, is mass, is the initial velocity, and is the final velocity. The most common mistake people make is forgetting that velocity has a direction. Your first step in any problem should be to define a positive direction (e.g., to the right). Any velocity going in the opposite direction must be given a negative sign. If an after-velocity is unknown, write it as and let the sign of your answer tell you the direction. Get this right, and you're halfway to the solution.
A special case you'll see is coalescence. This is a fancy word for when the particles stick together after impact, like two pieces of clay smooshing into one. They move off with a single, common final velocity, which we'll call . This actually simplifies our equation to:
Kinetic energy is not conserved. Momentum is always conserved in a collision, but some kinetic energy turns into heat, sound and denting. To find the loss, use momentum first to find every velocity after the impact, then work out total KE before total KE after, with for each body. The loss should come out positive. Never write "KE before = KE after" to find a velocity.
Multi-stage problems. Paper 4 often puts a collision in the middle of a longer story: a particle slides down a slope, hits another, and the combined particle then slows on a rough surface. Split it into motion stages (constant-acceleration formulae, or energy) and impact stages (conservation of momentum), keep one positive direction throughout, and use momentum conservation only across an impact, never during the sliding or falling. Two particles collide again only if the one behind is moving faster, in the same direction, than the one in front.
Impulse, the coefficient of restitution, explosions and collisions at an angle are not in the 9709 syllabus, so you never need them for Paper 4.
Worked example
Worked Example: Direct Impact with Coalescence
The 'We're Sticking Together' Problem 🥰
A particle A of mass is moving with a velocity of on a smooth horizontal surface. It collides directly with a particle B of mass which is moving in the opposite direction with a velocity of . After the collision, the two particles coalesce to form a single particle. (a) Find the velocity of the combined particle after the collision. (b) Find the loss in kinetic energy due to the collision.
- 1First things first, let's set up our system. We must define a positive direction. Let's take the initial direction of particle A (to the right) as positive. This means any velocity in the opposite direction will be negative.Positive direction:
For Particle A: ,
For Particle B: , - 2The problem states the particles coalesce, so we'll use the specific version of the conservation of momentum formula for this scenario. This shows the examiner you've correctly identified the type of collision.
- 3Now, we substitute our known values into the equation. Be super careful with the signs: this is where mistakes happen!
- 4Time to simplify and solve for , the common final velocity. This is just algebra from here on out.
- 5We state our answer clearly with units and interpret the result. Since our value for is positive (), the combined particle moves in the direction we initially defined as positive (the original direction of particle A). It also lies between the two starting velocities, as a coalescence answer always does.(a) The velocity of the combined particle is in the original direction of motion of particle A.
- 6For part (b), use the velocities we now know. Squaring removes the signs, so kinetic energy is never negative.
Answer
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