In mechanics we find that Newton’s second law can’t be applied when certain forces are involved. We can, however, introduce two concepts so that we may apply equations to solve problems we couldn’t before. As long as we stay within the domain of Newtonian mechanics the concept of conservation of momentum will help us analyze situations where Newton’s laws would prove difficult to use.
Here, we will replace Newton’s second law with momentum and impulse.
First let’s derive our momentum equation using Newton’s second law. We’ll consider a particle with mass m and remember that acceleration is the time derivative of velocity. That gives usand since our mass is constant we can take m inside the derivativeNow Newton’s second law states that the net force on a particle is equal to the time rate change of mv, which we will call linear momentum, to be specific. Using the symbol p to denote momentum we finally have our equation Important things to note is that momentum is a vector quantity where its magnitude is (mv) and direction id v.
This means that the momentum of a baseball thrown east is different than that same baseball thrown west because of their directions. The momentum of a truck driving 20 m/s is different than a car diving 20 m/s because of their masses. The momentum of a professional soccer player kicking a soccer ball is different than a child kicking the same ball because of their speeds.
Because it’s fun to manipulate equations (and it will prove useful later) we will rewrite our momentum equation in terms of Newton’s second law. It’s interesting to know that originally Newton stated the second law as the net force acting on a particle equals the time rate change of the quantity of motion of a particle.
Perhaps not as interesting, but vital to note is that the law is only valid in inertial frames of reference.
Now we’re going to explore momentum and kinetic energy to then define impulse.
While momentum of a particle is a vector quantity dependent velocity, the kinetic energy of a particle is a scalar quantity dependent on velocity squared. This is the mathematical difference between momentum and kinetic energy, but to see the physical difference we must introduce impulse.
To do this we will introduce the symbol J for impulse while considering a particle acted on by a constant force F during a time interval Δt. That is Like momentum, impulse is a vector quantity. To understand what we use impulse for we’ll go back to our momentum equation and manipulate it into the impulse equation. I will do the deriving quickly to save timeWe’ve now arrived at the impulse-momentum theorem which states that the change in momentum of a particle during a time interval equals the impulse of the net force that acts on the particle during that interval. This theorem will also hold even when forces aren’t constant. In order to show that this is true we take integrate both sides of Newton’s second law written in terms of momentum. I will also do this derivation all at once.
We can also use the impulse-momentum theorem when the net force is not constant by defining an average net force. To do so we use the equation
Perhaps the most important thing to remember about momentum and impulse is that they are both vector quantities. This allows us to simplify our equations when solving problems by using the equations in their component forms of (x,y,z).
Now all that deriving wasn’t for fun (it was fun though!). What we did through derivation was to show the physical difference between momentum and kinetic energy I previously mentioned.
The impulse-momentum theorem states that the changes in a particle’s momentum are due to impulse; depends on time when the net force acts. The work-energy theorem states that the kinetic energy changes when work is done on a particle; depends on the total work done over the distance the net force acts.
To further analyze this let’s consider a particle that starts from rest. Its initial momentum is zero and the kinetic energy is zero. Next we’ll introduce a constant net force F to act on the particle from time t1 to t2 , moving it a distance s in the direction of F. Using our simple impulse equation (J= p2-p1) at the final time we have
That confusing equation states that the momentum of a particle equals the impulse that accelerated it from rest to its present speed. Comparing this to kinetic energy we see that the work done at the final time is which is the total work done to accelerate the particle from rest.
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