The formula for force is
F = ma
The formula for the force of gravity is
The force of gravity between the earth and a body on its surface is the weight of the body.Isaac Newton came up with these two formulas in 1687.The force acting on a falling body is the force of gravity. For a falling body, we have
F = Fg
The formula for kinetic energy is
KE = ½mv²
For a body that starts from rest and travels with constant acceleration, we have the following formula
v² = 2ad
Multiplying by ½m
½mv² = madKE = mad
Differentiating this formula
DKE = ma
The derivative of kinetic energy, with respect to distance, is mass times acceleration.
A body weighs less when it is moving.The rest weight of a body is how much it weighs when it isn’t moving.The formula for rest weight is
Weight is equal to rest weight minus kinetic energy.The rest weight of a falling body is increasing.The kinetic energy of a falling body is increasing.The weight of a falling body stays the same.
The derivative of a constant is zero.For a falling body, the derivative of weight is zero.
The derivative of rest weight is
The formula for weight is
W = RW − KE
Differentiating this formula
DW = DRW − DKE
The positive direction is up. The acceleration of a falling body is downward, so the acceleration is negative.
A block is on a flat, frictionless surface. The block is attached to a spring. The other end of the spring is attached to a wall. The block is pulled out a short distance. This stretches the spring. The block is released. The spring starts to contract, pulling the block back toward the wall. What is the acceleration of the block?
The formula for spring force is
Fs = −kx
The force acting on the block is the spring force of the spring. We have
F = Fsma = −kxa = −kx/m
The positive direction is away from the wall. The block is pulled toward the wall, so the acceleration is negative.
My Way
A stretched spring has energy. The energy is known as elastic potential energy. The formula for elastic potential energy is
EPE = ½kx²
The derivative of elastic potential energy is
DEPE = kx
The block has kinetic energy.The total energy of the spring and the block stays the same.The derivative of total energy is zero.
For the spring and the block, we have
TE = EPE + KE
Differentiating this formula
DTE = DEPE + DKE
0 = kx + maa = −kx/m
If a one-kilogram body falls ten centimeters near the surface of the earth, the kinetic energy of the body goes up one joule. What else changes?
Helmholtz's Way
Gravitational potential energy is the integral of the force of gravity. The formula for gravitational potential energy is
Hermann Helmholtz came up with this formula in 1847.The value of GM for the earth is 399 trillion meters cubed per seconds squared.The mean radius of the earth is 6,371,000 meters.The gravitational potential energy of a one-kilogram body on the surface of the earth is −62,627,531 joules.The gravitational potential energy of a one-kilogram body at a height of ten centimeters is −62,627,530 joules.If a one-kilogram body falls ten centimeters near the surface of the earth, gravitational potential energy goes down one joule.Mechanical energy is equal to kinetic energy plus gravitational potential energy.For a falling body, kinetic energy goes up, gravitational potential energy goes down, and mechanical energy stays the same.
My Way
The rest weight of a one-kilogram body on the surface of the earth is 62,627,531 joules.The rest weight of a one-kilogram body at a height of ten centimeters is 62,627,530 joules.If a one-kilogram body falls ten centimeters near the surface of the earth, rest weight goes up one joule.For a falling body, rest weight goes up, kinetic energy goes up, and weight stays the same.
What is the formula for escape velocity?
Their Way
A space probe slows down as it recedes from the earth.The velocity of a space probe at an infinite distance is zero, so the kinetic energy of a space probe at an infinite distance is zero.The gravitational potential energy of a space probe at an infinite distance is zero.The mechanical energy of a space probe at an infinite distance is zero.The mechanical energy of a space probe stays the same once the probe is launched. A space probe escapes from the earth if its mechanical energy when it is launched is zero. The formula for escape velocity is
ME = KE + U0 = KE + UKE = –U
My Way
If a body is moving fast enough, its weight will be zero. There are then no gravity bonds holding the body to the earth and the body escapes. The formula for escape velocity is
W = RW – KE0 = RW – KEKE = RW
The formula for the velocity of a satellite in a circular orbit is
Multiplying by m
Their Way
The gravitational potential energy of a satellite in a circular orbit is
U = –2KE
The mechanical energy of a satellite in a circular orbit is
ME = KE + UME = KE – 2KEME = −KE
My Way
The rest weight of a satellite in a circular orbit is
RW = 2KE
The weight of a satellite in a circular orbit is
W = RW − KEW = 2KE − KEW = KE
The orbit of a satellite around a primary is shaped like an ellipse, with the primary located at one focus of the ellipse.The point in its orbit where a satellite is closest to the primary is known as periapsis. The point in its orbit where a satellite is farthest from the primary is known as apoapsis. Periapsis and apoapsis are located at the end points of the major axis of the orbit.The velocity of a satellite is changing. The closer a satellite is to the primary, the faster it is traveling.
Their Way
What is the mechanical energy of a satellite in an elliptical orbit?
For a satellite in an elliptical orbit, we have the following formula
vA = VP
v is the velocity of a satellite at apoapsis, A is the distance of a satellite from the primary when the satellite is at apoapsis, V is the velocity of a satellite at periapsis, P is the distance of a satellite from the primary when the satellite is at periapsis.Squaring this formula and multiplying by ½m
½mv²A² = ½mV²P²
The mechanical energy of a satellite in an elliptical orbit stays the same. We have the following formulas
Multiplying the first formula by A squared and the second formula by P squared
Subtracting the second formula from the first formula
ME A² − ME P² = GMmP – GMmA
Factoring
ME (A² − P²) = GMm (P −A)ME (A + P) (A – P) = −GMm (A – P)ME (A + P) = −GMm
A and P together make up the whole major axis of the orbit. The length of the major axis of an ellipse is 2a. We have
ME 2a = −GMmME = −GMm/2a
My Way
What is the weight of a satellite in an elliptical orbit?The kinetic energy of a satellite in an elliptical orbit goes up and down, but the weight of the satellite stays the same. Part of the time, kinetic energy is greater than weight, and part of the time, kinetic energy is less than weight.There are two points in a satellite’s orbit where kinetic energy is equal to weight. These two points are the two end points of the minor axis.
We have the following formulas for a satellite when it is at one of the end points of the minor axis
W = RW – KEW = KE
Combining these two formulas
W = RE – W2W = RWW = ½RW
The distance from one of the foci of an ellipse to one of the end points of the minor axis is a. The rest weight of a satellite at one of the end points of the minor axis is
RW = GMm/a
The weight of a satellite at one of the end points of the minor axis is
W = GMm/2a
This is the weight of a satellite at any point in its orbit.
These formulas are all wrong
F = ma
Fs = −kx
These are the correct formulas
DKE = ma
DEPE = kx
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