Momentum Review

Hello readers,

Momentum-
  1. a property of a moving body that the body has by virtue of its mass and motion and that is equal to the product of the body's mass and velocity; broadly :  a property of a moving body that determines the length of time required to bring it to rest when under the action of a constant force or moment
--Webster definition

Momentum=mass (kg)*velocity(m/s)
                 p =m*v
Whenever we write momentum, we must remember that it is on a vector, meaning we are required to indicate the direction, in both a sketch, which will say which way is positive and negative, and in our equations through the use of - to indicate going down or backwards on the x and y scale.

Any object in motion has momentum, however if it is at rest, it has none. This is clear on a free body diagram, where despite an object having a force due to gravity, and a normal force, if it is stationary, there is none due to momentum. The unit of momentum is kg per m/s.

Sample problem:
A bowling ball weighing 8 Kg is rolling along at 2 m/s, what is its momentum?
      p=m*v
      p=8*2
      p=16kg/(m/s)
Momentum = 16Kg/(m/s)
The next step in momentum is impulse. Impulse is derived from Newton's second law, which states that:
The acceleration of an object as produced by a net force is directly proportional to the magnitude of the net force, in the same direction as the net force, and inversely proportional to the mass of the object.
--Physics classroom 
Mass is always measured in Kg.
Acceleration is: (Δv/ΔT)
 Forcenet=mass*acceleration
 wherein force and acceleration are directly proportional, and mass and acceleration are inversely proportional.
 This means that if acceleration increases, so does force directly. And if the force stays the same, but the mass increases, acceleration decreases in a direct manner.

 The unit of impulse (J) is newtons, and it is the force times the time. Impulse=change in momentum.
 Thus we have the formula:
Impulse=(forcenet*change in time=(mass*acceleration))*change in time.)
Thus, j=p
which means fnet*Δt=m*(a=(Δv/Δt)).
 So our final impulse formula is:
j=fnet*Δt

 Sample problem:
What is the impulse of a 1800 kg car that collides with a fencepost weighing 20 Kg for 3 seconds?
First, to find fnet I multiply 1800*10 which is 18000
-----------> j=fnet*Δt
                  j= 18000*3
                  j=54000N

Now that we know what momentum and impulse are, we can learn the new formula's which take advantage of this understanding in collision. However there is a mathematical model that we base these formula's on.
This is p(total before)=p(total after) as energy must be accounted for throughout.
There are 2 types of collisions:
Unstuck--> unstuck  =  mava+mbvb=mava+mbvb
 Unstuck-->stuck      =  mava+mbvb=(ma+mb)*(Va*b)

We must remember that the force in which two objects hit each other is always equal due to newton's third law, however the repercussions on each other in terms of momentum can be different.

Sample problem:
Unstuck to stuck
If a 150 kg skateboarder travelling at 15 m/s hits a 275 kg marble ball at rest, what is the final velocity of the two travelling forward after they collide?
mava+mbvb=(ma+mb)*(Va*b)
150*15+275*0=(150+275)*(vab)
2250= 475*Vab
2250/475= (475*Vab)/475
Vab=4.7 m/s.

Unstuck to Unstuck 
A 2500 kg blue car is travelling forward at 40m/s, and collides with another 1500 kg red car traveling 16 m/s, what is the red car's speed after rebounding off each other if the blue car slows down to 8 m/s?
Formula: mava+mbvb=mava+mbvb
 (2500*40)+(1500*16)=(2500*8)+(1500*vb)
                         124000=20000+1500VB
              124000-20000=20000+1500VB-20000
                         104000=1500Vb
                104000/1500=1500Vb/1500
                     69.33 m/s=vb
The red car speeds up to 69.33 m/s.

Using energy to predict velocity

This lab was designed to see if we could cause two carts to travel at the same velocity while only being given the spring constant of both springs (red and blue).
We labeled our data according to spring color.

The two formula's we used were:
Ek=1/2*m*v^2
Eel=1/2Kx^2

Red:
K=84
M= 0.55 Kg

Blue:
K=112
M= 0.5422 Kg


Once we had this data, we set a compression to red, of 5 cm, 0.05 M, and then input it into the formula to solve for energy:
Eel=1/2Kx^2
Red=0.5*84*0.05^2
       = 0.105j
and then we set the energy equal to solve for the Blue compression:
blue=0.5*112*x^2
0.105=56x^2
x= 0.043m

To then solve for velocity, we input this data into the formula:
Eel=Ek because inital energy= final energy
so
Both red and blue Eel=0.105 j which means 0.105j=EK
Thus to solve for velocity we input the energy (0.105j) into the Ek position.
 0.105j=1/2*m*v^2
Red:
0.105j=0.5*0.55*v^2
0.105j=0.275*v^2
0.105j/0.275=(0.275*v^2)/0.275
0.381=v^2
v= 0.617m/s

Blue:
0.105j=0.5*0.5422*v^2
0.105j=0.2711*v^2
0.105j/0.2711=(0.2711*v^2)/0.2711
0.387=v^2
       v=0.622m/s


To double check our blue compression data, we input the velocity of red (0.61 m/s) into the formula to find our Ek:

Ek=0.5*0.5422*0.617^2
Ek=.103j

Which when used as part of the Eel=Ek formula means that Eel=.101.
Thus, when we input this into the Eel formula, we solved for the compression of x for blue.
 Eel=1/2*k*x^2
.103=0.5*112*x^2
.103=56*x^2
.103/56=(56*x^2)/56
 .0018=x^2
 0.0424=x

When we tested the carts, we pulled the cart with the red spring 0.05m (5 cm) back, and the cart with the blue spring back 0.042m (4.2 cm).
We used pasco as our testing software, and our red cart came in at 0.56 m/s for velocity, while the blue cart came in at 0.61 m/s for velocity.
This resulted in a percentage of error for our red cart of (error %=(0.61-0.56)/0.61) 8.1% error.

Energy Transfer model

Welcome, one and all to energy transfer principles.

The formula's used are
Ek: 1/2 m*v^2
Eel: 1/2kx^2
Eg: m*g*h
W: fnet*deltax

Across charts, we use the idea that energy is a closed system, except for Ediss, where once it disappears to, it is gone.
This results in formula's such as
Ek=Eg, where one variable needs to be solved for, or a time concept, or a force present can be used.

The other concept is joules, these principles of energy that when indicated by time(seconds) become watts, and in hours kwhr's.
Across energy transfer, we can use pie charts which are conceptual, to LOL charts which are more precise. One of the necessary factors to remember is that in the system which defines the energy entering or leaving, if we are using earth's gravitational force, it must be included.

If we have a 150 kg skateboarder going at 3 m/s, when friction is negligible, what is his Ek?
Ek=1/2 m*v^2
     = 0.5*150*3^2
     = 675 j

This same skateboarder accidentally goes off a cliff whose height we do not know.
To solve for Height we can use:
Ek=Eg
675j=M*g*h
675j=150*10*h
675j/1500=1500h/1500
h= 0.45 M.

If a spring with a constant K of 115 has 280 j of energy stored, how far was it compressed?
  Eel=1/2kx^2
280j= 0.5*115*x^2
280j=57.5x^2
280j/57.5=(57.5x^2)/57.5
4.87=x^2
x=2.2 m