Let us look at the principle of impulse and momentum now to do this, we need to recap just quickly
the principle of work and energy and see how that is different from the principle of impulse and momentum.
So the principle of work and energy, we could solve problems involving forces, positions and speed.
So to get from velocity one to velocity, too, we considered the work done on a body to give it velocity
to, and that work was a force that worked in over a certain distance.
Now, with Impulse, we are going to look at problems involving forces, time and velocity.
So to get from velocity one to velocity too, we apply a certain impulse, which is a force that works
in over a certain time.
So work is a force that works in over a certain distance.
Impulse is a force that works over a certain time.
So that is more or less the difference between the two.
And a force can create an impulse on a particle without doing work on that particle.
So even if the particle or the body is not moving, they can still be an impulse, whereas with work
there is always displacement involved.
So, as I've said, an impulse is basically a force multiplied by the time in which that force works
in.
So let us first look at the case for constant force.
We know that according to Newton's second law, F is equal to M multiplied by eight.
So if I multiply both sides with T, I get on the left and side.
If T is equal to M80.
Now, according to our equations for Motion, V1 is V0 plus 80, so therefore 80 is V1 minus V0.
So I can substitute that in for 80 and then we get if this is equal to M V1 minus V0 and F.T. Force
multiplied by the time that is called the impulse.
And so the impulse is equal to M.V. one minus in V zero or you could say in two minus M.V. one.
But it comes down to the fact that it is an initial velocity and a final velocity.
Before and after the impulse, so for a variable force, we need to consider the time period for each
magnitude of the force.
If there's a variable force that varies in time, we need to integrate.
So we say the some of the forces and I, I make this force a vector, it's got direction is equal to
M.A that is equal to MDVIP.
And if we multiply both sides by DETI, we get if it is in the V now, we can integrate.
Now remember, this force is variable.
That is why we need to integrate between our starting time to one and our ending time two.
So the integral of FTD between T1 and T2 is equal to the integral between velocity and velocity of MTV
exactly the same as what we did in the previous slide.
This is just with a variable force and therefore I is equal to M V to minus V1.
No I r impulse that is the integral between T one and two of FTT.
So once again it's just multiplying the force with the time and because the forces variable we're going
to integrate.
So we're going to sum all these different little components of the force multiplied by the time.
And so the impulse is equal to M.V. to minus one and we can rearrange that to say, M.V. one plus the
impulse is M.V. two, and that is our principle of impulse and momentum.
So you can see that you have an M.V. one or a momentum.
If you add to that an impulse, you're going to get a second new momentum, M.V. two.
And so because impulse is force multiplied by time, if you plot the force as a function of time, the
area under the force time curve, that will give you your impulse.
So the larger the force, the larger the impulse, the larger the time, the larger the impulse, because
it's the area under the force time curve.
Now, what I've drawn here is a variable force.
That is why we need to integrate.
If it was a constant force, it would have just been a straight line and we could just take if multiplied
by t to get the area under the curve, as we've done in the previous slide.
So it's the same principle.
But you should just remember for constant force, you can just take if multiplied by T and for a variable
force, you'll have to integrate.
Now let us illustrate impulse and momentum in two in three different ways, actually.
So the equation says M.V. one plus the impulse is equal to M.V. two.
Now I have written this in vector form because the velocities and the impulses can be broken down in
the three Briem primary directions.
We can draw it as a vector diagram, so if M.V. one is to the right and the top and we add an impulse
that goes to the top and to the left, then M.V. two will be to the left and the top.
So M.V. one plus the impulse will give you M.V. to we can draw in terms of using balls.
So if you have a ball drawn here by a blue circle with a momentum in one going to the right and the
top, plus an impulse on the ball that goes to the left and the top, that would give you a velocity
multiplied by a mass going to the left and the top.
So M.V., that is momentum.
And you'll recognize this also from from school.
So let us break down M.V. one plus I'm equals M.V. two into its different components.
So predominant as a vector.
Let's break it down into its X, Y and Z components.
So if we expand that equation, it'll say M.V. one in the X direction.
Plus the impulse in the X direction is in between the X direction.
Similarly, in the one in the Y direction, plus the impulse in the Y direction is M.V. two in the Y
direction.
Similarly, M.V. one in the Z direction.
Plus the impulse in the direction is M.V. two in this direction.
So the impulse has components and we can find the magnitude.
How large is this impulse?
We get that by taking the square root of the sum of the squares of the different components of the impulse.
So the magnitude of the impulse is a square root of the impulse in the X direction squared, plus the
impulse in the Y direction squared, plus the impulse in the in the Z direction squared.
So this is the theory you need to know to apply the principle of impulse and momentum on a basic problem.
So in the next video, we'll have a look at an example.