Passing The Class

Law Of Conservation Of Energy- is the total energy of an isolated system that is constant

Mathematical Representation
Abbreviated Version- Ei= Ef is the shorter version of the extended
Extended Version- KEj + PEj + W nc + OEj = KEf + PEf + OEf

Different Units Of Energy
SI Units- SI units are based on Joules
American Units- American units are MKS units of energy which is equal to one newton.

Unique Video- https://www.youtube.com/watch?v=LrRdKmjhOgw

In the video of the coaster it is shown how that the roller coaster has potential energy. When it is at the top of the hill it is just being pulled up and stops for a quick moment and drops the energy is released. The time it is moving it is called kinetic energy because it is moving down the hill to complete the ride. By the time the ride is over all of the kinetic energy is slowly decreasing because the roller coaster needs the energy to get around the ride.

Question
A cart travels along a frictionless roller coaster track. At point A, the cart is 10 m above the ground and traveling at 2 m/s.

A) What is the velocity at point B when the cart reaches the ground?

B) What is the velocity of the cart at point C when the cart reaches a height of 3 m?

C) What is the maximum height the cart can reach before the cart stops?

Solution:

The total energy of the cart is expressed by the sum of its potential energy and its kinetic energy.

Potential energy of an object in a gravitational field is expressed by the formula

PE = mgh

where
PE is the potential energy
m is the mass of the object
g is the acceleration due to gravity = 9.8 m/s2
h is the height above the measured surface.

Kinetic energy is the energy of the object in motion. It is expressed by the formula

KE = ½mv2

where
KE is the kinetic energy
m is the mass of the object
v is the velocity of the object.

The total energy of the system is conserved at any point of the system. The total energy is the sum of the potential energy and the kinetic energy.

Total E = KE + PE

To find the velocity or position, we need to find this total energy. At point A, we know both the velocity and the position of the cart.

Total E = KE + PE
Total E = ½mv2 + mgh
Total E = ½m(2 m/s)2 + m(9.8 m/s2)(10 m)
Total E = ½m(4 m2/s2) + m(98 m2/s2)
Total E = m(2 m2/s2) + m(98 m2/s2)
Total E = m(100 m2/s2)

We can leave the mass value as it appears for now. As we complete each part, you will see what happens to this variable.

Part A:

The cart is at ground level at point B, so h = 0 m.

Total E = ½mv2 + mgh
Total E = ½mv2 + mg(0 m)
Total E = ½mv2 

All of the energy at this point is kinetic energy. Since total energy is conserved, the total energy at point B is the same as the total energy at point A.

Total E at A = Total Energy at B
m(100 m2/s2) = ½mv2

Divide both sides by m
100 m2/s2 = ½v2

Multiply both sides by 2
200 m2/s2 = v2

v = 14.1 m/s

The velocity at point B is 14.1 m/s.

Part B:

At point C, we know only a value for h (h = 3 m).

Total E = ½mv2 + mgh
Total E = ½mv2 + mg(3 m)

As before, the total energy is conserved. Total energy at A = total energy at C.

m(100 m2/s2) = ½mv2 + m(9.8 m/s2)(3 m)
m(100 m2/s2) = ½mv2 + m(29.4 m2/s2)

Divide both sides by m

100 m2/s2 = ½v2 + 29.4 m2/s2
½v2 = (100 – 29.4) m2/s2
½v2 = 70.6 m2/s2
v2 = 141.2 m2/s2
v = 11.9 m/s

The velocity at point C is 11.9 m/s.

Part C:

The cart will reach its maximum height when the cart stops or v = 0 m/s.

Total E = ½mv2 + mgh
Total E = ½m(0 m/s)2 + mgh
Total E = mgh

Since total energy is conserved, the total energy at point A is the same as the total energy at point D.

m(100 m2/s2) = mgh

Divide both sides by m

100 m2/s2 = gh

100 m2/s2 = (9.8 m/s2) h

h = 10.2 m

The maximum height of the cart is 10.2 m.

Answers:

A) The velocity of the cart at ground level is 14.1 m/s.
B) The velocity of the cart at a height of 3 m is 11.9 m/s.
C) The maximum height of the cart is 10.2 m.

Part 2

                 Jose with a mass of 85 kg, went bike riding early in the morning, He comes to rest to check how many miles has he traveled. He comes to a stop 15 m hill. From rest Jose starts pedaling and goes down the hill. How fast is Jose going at the bottom of the hill?

SOLUTION
                       Know                            Unknown                                      Work/Answer
85 Kg                                                   Vi=?                                GPE= m * g * h
15 m                                                                                             GPE= 85 * 9.8 * 15
                                                                                                     GPE= 833 * 15
                                                                                                     GPE= 12495

3 SCREENSHOTS
                             

Sentence

The graph looks like this because it is showing how high and how fast it would be going  on the ramp.

Sentence

The graph is basically how the rider would it look when the rider is upside down on the loop.

Sentence

This screenshot of where the rider has the most speed when he is on the loop.