84E - Momentum

In classical mechanics, momentum (pl. momenta; SI unit kg m/s, or equivalently, Ns) is the product of the mass and velocity of an object. For example, a heavy truck moving fast has a large momentum - it takes a large and prolonged force to get the truck up to this speed and it takes a large and prolonged force to bring it to a stop afterwards. If the truck were lighter, or moving more slowly, then it would have less momentum.

Like velocity, momentum is a vector quantity, possessing a direction as well as a magnitude.

Momentum is also a conserved quantity, meaning that if a closed system is not affected by external forces, its total momentum cannot change. In classical mechanics, conservation of momentum is implied by Newton's laws.

84E1 - Momentum and Impulse

Here you will learn to:

Define momentum and solve problems using p=mv,

Define impulse and solve problems using Ft = Δp,

Define momentum and impulse in terms of Newton's second law, and

Describe situations where impulse is used to reduce the risk of injury.

Momentum

Momentum is a vector quantity in physics with the symbol p and units of newton seconds (Ns). It is defined as the product of mass and velocity (p = mv). A car of mass 1000 kg travelling along a freeway at 30 ms-1 has a momentum of 30 000 Ns (1000 x 30).

What does this mean though in a practical sense? Momentum is related to the force needed to stop an object in a certain period of time. The larger the mass and velocity, the greater the force needed to stop the object. Another car with the same mass and moving at 3 ms-1 has a momentum of
3000 Ns (1000 x 3). This means that ten times more force is needed to stop the faster car in the same period of time because it has momentum that is ten times greater than the slower car.

Let's look at another example to highlight how both mass and velocity contribute to momentum. Consider again our car of mass 1000 kg travelling along a freeway at 30 ms-1 with total momentum equal to 30 000 Ns. How fast would a 1 kg book need to be thrown in order to have the same momentum as the car? Because the mass of the book is 1000 times less than the mass of the car, the velocity of the book must be 1000 times greater than the velocity of the car. If the book was thrown with a velocity of 30 000 ms-1, it would have the same momentum as the car and need the same force to stop it in the same period of time.

Impulse

Momentum can also be described in terms of Newton's Second Law if we measure the momentum of an object over a small period of time. Newton's Second Law can also be stated as 'the applied force is directly proportional to the time rate of change of momentum'. Taking the formula F = ma and substituting a = (v-u)/t we get a mathematical expression F = m(v-u)/t. This tells us that their is a linear relationship between the force and the change in momentum for any object if the time interval for the change in momentum is the same. This means that to accelerate a 1000 kg car from rest to a speed of 30 ms-1 (p = 30 000 Ns) in 1 second we need a net force of 30 000 N. If it happened in 10 seconds then we would only need a force of 3000 N and if it took 100 s we would only need 300 N. The same could be said for the force needed to stop the car but it would just be acting in the opposite direction.

As we have seen, the applied force is equal to the time rate of change of momentum (Newton's second law). Air bags in cars, bicycle helmets and running shoes are all good examples of this physics principle put to good use in our everyday lives. Running shoes are designed to minimise the impact of a collision with a hard surface and usually contain compressed air or some other compressible substance. Wearing these shoes increases the time it takes for a runner's foot to come to rest when it comes into contact with the ground because of the time it takes to compress the heel. Increasing time in the denominator of the expression on the right decreases the applied force and this minimises the shock of the collision with the ground. The same goes for air bags and bicycle helmets. Increasing the stopping time decreases the force and protects the passengers from injury.

Describing Newton's second law in terms of momentum (right) also provides another useful quantity in physics called impulse. Impulse is the product of the applied force and the time over which it acts and is equivalent to the change in momentum of the object. Consider a tennis ball being hit by a racquet. A large force and a long follow through with the racquet will give the largest impulse and the greatest change in momentum. Since change in momentum is m(v-u), a large change in momentum means that the tennis ball has a large change in velocity and leaves the racquet with a high velocity. This is why tennis players are told to hit the ball hard and long in a serve to give it the largest change in momentum, and hence velocity, possible.

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84E2 - Conservation of Momentum

In this section you will learn to:

Apply the law of conservation of momentum to solve problems involving collisions and other interactions.

The Law of Conservation of Momentum

From Newton's third law of motion we know that whenever a force is appliedby an object to another object, there will be an equal and opposite reaction force applied to the object applying the force in the first place. In terms of a car crash, when Car A crashes into Car B, the force of Car A on Car B is equal and opposite to the force of Car B on Car A. Action and reaction forces cause objects to accelerate in opposite directions. If Car B is stationary, it will accelerate forwards while Car A will come to rest because of an acceleration in the other direction (deceleration).

In any collision like this the sum of initial momentum of the two cars in the system before collision is found to be equal to the sum of the final momentum of the cars after collision. Thus Newton's second and third laws of motion lead us to the very important law of mechanics, the law of conservation of momentum. The the law of conservation of momentum states that if a group of bodies are exerting a force on each other (the system) then their total momentum remains the same before and after the collision provided there is no external force acting on them. The same goes for the opposite of a collision, such as an explosion. The sum of the momentum of a bomb ball before exploding is equal to the sum of the momentum of each part moving away from each other after the explosion.

Conservation of momentum in a collision between two cars of the same mass (1000 kg).

So what does conservation of momentum mean in reality? Consider a situation like the one shown on the right. Car A of mass 1000 kg is moving at a speed of 15 ms-1 to the right and Car B with the same mass is stationary. The law of conservation of momentum tells us that the total momentum of Car A and B before the collision must be equal to the total momentum of Car A and Car B after the collision. In other words, the momentum lost by Car A is the same as the momentum gained by Car B. The momentum before the collision is equal to the momentum of Car A (15 000 Ns) plus the momentum of Car B (0 Ns). The total momentum before the collision is 15 000 Ns in the forward direction. From the law of conservation of momentum, the total momentum after the collision must also be 15 000 Ns, so if Car A comes to rest in the collision, Car B must move off with a momentum of 15 000 Ns and a velocity of 15 ms-1 in the forward direction.

Collisions

The law of conservation of momentum applies is every interaction between objects, however, kinetic energy is not always conserved in the same way. Considering our example above, some of the kinetic energy of Car A will be converted in heat and sound as well as work done to deform the other car. While all of the momentum of Car A is transferred to Car B, not all of Car A's kinetic energy is transferred to Car B. Collisions where kinetic energy is not conserved are called inelastic collisions. On the other hand, elastic collisions where kinetic energy is conserved are rare in nature and generally only occur between the molecules of somewhat idealised gases.

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