M1 Mechanics

Momentum

Momentum is the quantity of motion, and is a product of the mass and velocity of an object.

Momentum = mass x velocity

Where p is momentum in kgms^-1, m is mass in kg and v is velocity in ms^-1

Momentum and velocity are vectors because they have both a magnitude (size) and a direction. Mass is a scalar quantity because it has no direction.

Conservation of momentum

Imagine this: we’re on the rink at an ice hockey game. There’s no friction worth worrying about. Two players skate towards each other and collide. One bloke weighs a lot more than the other, but the lighter guy skates much faster.

But what happens when they collide? Their momentums combine. If the speeds are exactly right, the slow and heavy bloke will have exactly the same momentum as the light, fast dude. If they’re going straight at each other, their momentums will cancel each other out as they collide. That means they’ll both come to a stop as they crash.

If things were different, the heavy bloke might be going the same speed as the little guy. When they collide this time, the heavier lad will have a lot more momentum because he has a lot more mass. What a porker! Anyway the result of his bulkiness would be that he slows down a bit, but he sends the lighter player flying. Ouch!

In general, if we call the really lardy bloke A and we call the lighter guy B, we can rehash the p = mv equation to make this:

m_Au_A + m_Bu_B = m_Av_A + m_Bv_B

Where m_A is the mass (in kg) of particle A (the chunkster), u_A is particle A’s velocity (speed in a certain direction in ms^-1) before the collision and v_A is his speed after the crash. You can rearrange that equation to find all sorts of things.

Often in the exam you’ll be given m_A, u_A, v_A, m_B and u_B and you’ll have to find v_B. You do it like this:

m_Au_A + m_Bu_B = m_Av_A + m_Bv_B

→ (m_Au_A + m_Bu_B - m_Av_A) / m_B = v_B

Newton's Laws of Motion

Superstar English scientist and mathematician Sir Isaac Newton discovered three rules which describe motion. So famous did he become, Cambridge University named a whole building after him. The M1 module is entirely based on so-called Newtonian mechanics invented by the man himself. Check out those curls - he may have kick-started modern physics but he sure didn’t have a clue about haircuts! Looks like Justin Hawkins from The Darkness.

Newton's First Law of Motion:

I. Every object in a state of uniform motion tends to remain in that state of motion unless an external force is applied to it.

This is often termed simply the "Law of Inertia".

Newton's Second Law of Motion:

II. The relationship between an object's mass m, its acceleration a, and the applied force F is F = ma.

Acceleration and force are vectors (as indicated by their symbols being displayed in bold font); in this law the direction of the force vector is the same as the direction of the acceleration vector.

Newton's Third Law of Motion:

III. For every action there is an equal and opposite reaction.

Imagine what happens if you step off a boat onto the bank of a lake: as you move in the direction of the shore (towards it), the boat tends to move in the opposite direction (away from the shore). If you do this, you might fall in the water! But imagining it helps you remember that Newton’s Third Law of Motion applies to the real world.

Projectile Motion

What happens when Jonny Wilkinson gets anywhere near rugby ball? The ball goes flying towards anything H shaped nearby and lands shortly afterwards. It just so happens that all journeys like this have more or less the same shape, called a parabola.

Parabolas all have this curved shape. Rugby balls, cannon balls, missiles and paper aeroplanes all take this shape of path because they are under the influence… of gravity. When Jonny kicks a ball, his foot is going diagonally upwards. In mechanics you can consider his foot as going along and up at the same time, i.e. it has a horizontal and a vertical component of velocity.

Once he’s sent the ball flying, it too has alongwards and upwards velocity, but it is also having its upwards velocity constantly sapped away by gravity until it eventually starts to head back towards the ground. The alongwards velocity, however, is totally unaffected by what’s going on in the vertical direction.

Let’s look just at the horizontal component of the ball’s velocity first, since that bit’s the easiest. If Wilko kicks it with a certain force it will have a certain horizontal velocity. We’ll call that velocity v_x because x is the axis along which this horizontal velocity travels on a graph. The ball carries on with that velocity until it hits the ground. In mechanics you can totally ignore air resistance, largely because it doesn’t make very much difference, and also because it’s a pain in the arse.

Whatever’s going on vertically, this Australia-beating ball carries on horizontally, edging closer to the goal at a constant velocity, v_x.

If you want to find out how far it is from Jonny, all you need to know is v_x and the amount of time that has passed since he kicked it.

If the time is t seconds and the ball’s velocity is v_x ms^-1, the ball has travelled v_xt metres. I can say that because velocity = displacement / time, so displacement = velocity * time.

s_x = v_xt