Derive equation for a freely falling particles velocity

In summary, The equation that relates the velocity of a freely falling particle to its altitude is a=-g*[R^2/(R+y)^2], where R is the radius of the Earth and y is the altitude from the Earth's surface. The constant gravitational acceleration at sea level is denoted by g. The student attempted to derive the equation by taking the second derivative and using the chain rule, but got a different answer from what was shown. The topic being studied in class may be related to work, kinetic energy, potential energy, and related concepts.
  • #1
mathwizeguy
13
0

Homework Statement


Derive an equation that relates the velocity of a freely falling particle to its altitude. Assume that the particle is released from rest at an altitude from the Earth's surface.

Express your answer in terms of the variables and , constant gravitational acceleration at sea level , and the radius of the Earth .


Homework Equations



Given Equation :a=-g*[R^2/(R+y)^2]
R=radius of the earth.

The Attempt at a Solution


i tried taking the second derivative and used the chain rule but i got a different answer from what was shown.
 
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  • #2
mathwizeguy said:

Homework Statement


Derive an equation that relates the velocity of a freely falling particle to its altitude. Assume that the particle is released from rest at an altitude from the Earth's surface.

Express your answer in terms of the variables and , constant gravitational acceleration at sea level , and the radius of the Earth .


Homework Equations



Given Equation :a=-g*[R^2/(R+y)^2]
R=radius of the earth.

The Attempt at a Solution


i tried taking the second derivative and used the chain rule but i got a different answer from what was shown.

What, specifically, has your class been studying lately? Would it by any chance have something to do with work, kinetic energy, potential energy, and stuff like that?
 

1. What is the equation for a freely falling particle's velocity?

The equation for a freely falling particle's velocity is v = gt, where v is the velocity in meters per second (m/s), g is the acceleration due to gravity in meters per second squared (m/s^2), and t is the time in seconds (s).

2. How is the equation derived?

The equation is derived using the laws of motion and the constant acceleration formula, a = (vf - vi)/t, where a is the acceleration, vf is the final velocity, vi is the initial velocity, and t is the time. In the case of a freely falling particle, the initial velocity is 0 m/s and the final velocity is the velocity at a given time, so the equation becomes a = (v - 0)/t. Since the acceleration due to gravity is constant, it can be substituted for a, resulting in the equation v = gt.

3. What are the units of the equation?

The units of the equation are meters per second (m/s) for velocity, meters per second squared (m/s^2) for acceleration, and seconds (s) for time.

4. Can this equation be used for any freely falling object?

Yes, this equation can be used for any freely falling object as long as the acceleration due to gravity is constant. This includes objects such as a ball dropped from a height, a person jumping off a diving board, or a satellite orbiting the Earth.

5. How does air resistance affect the velocity of a freely falling object?

Air resistance can affect the velocity of a freely falling object by slowing it down. As the object falls through the air, it experiences air resistance which creates an upward force that opposes the force of gravity. This results in a decrease in the object's velocity compared to if it were falling in a vacuum with no air resistance.

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