Newton's Second Law of Motion: F=ma, Compute v^2

In summary, Newton's Second law of motion states that the force F acting on an object is equal to its mass m multiplied by its acceleration a. This law is accurate at low speeds but at high speeds, we must use the corresponding formula from Einstein's theory of relativity which takes into account the velocity function v(t) and the speed of light c. To simplify this equation to the traditional form of F=ma, we must ignore the ratio of v to c as it becomes negligible at low speeds. The derivative of this formula would be F=dp/dt=d(mv)/dt=m(dv/dt) which is not equivalent to F=ma.
  • #1
Jontafin410
2
0
Newton's Second law of motion F=ma, where m is the mass of the object that undergoes an acceleration a due to to an applied force F. This law is accurate at low speeds. At high speeds, we use the corresponding formula from Einstein's theory of relativity
F=m[tex]\frac{d}{dt}[/tex]([tex]\frac{v(t)}{\sqrt{1-\frac{v(t)}{c}}[/tex])

Where v(t) is the velocity function and c is the speed of light. Compute

v^2

What has to be "ignored" to simplify this expression to the acceleration a=v'(t) in Newton's second law?

 
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  • #2
do you mean
[tex]
F = m \frac{d}{dt}\frac{v(t)}{\sqrt{1-\frac{v(t)}{c}}}
[/tex]

what do you think?
 
  • #3
ignore the speed of light?
 
  • #4
At low speeds, is the ratio of v to c significant?

lanedance said:
do you mean
[tex]
F = m \frac{d}{dt}\frac{v(t)}{\sqrt{1-\left(\frac{v(t)}{c}\right)^2}}
[/tex]

what do you think?

Fixed it for you.
 
  • #5
no =) it such a small number it can be neglected. so ignore the ration. thank you
 
  • #6
Yep!
 
  • #7
Yea the equation thing on the sight is confusing. What would the derivative of that equation be?
 
  • #8
Well, firstly, I believe that it is [itex]F=ma\gamma[/itex], where [tex]\gamma=\frac{1}{\sqrt{1-\left(\frac{v}{c}\right)^2}}[/tex]. But I'm not certain.
 
  • #9
[tex]F=\frac{dp}{dt}=\frac{d(\gamma mv)}{dt}=m\frac{d}{dt}(\frac{v}{\sqrt{1-\frac{v^2}{c^2}}})}\neq\gamma ma[/tex]

You can't simply pull the gamma out of the derivative because it's v-dependent. You need to take the derivative. You can use the chain rule and the division rule.
 

1. What is Newton's Second Law of Motion?

Newton's Second Law of Motion states that the force acting on an object is equal to the mass of the object multiplied by its acceleration. In equation form, it is written as F=ma, where F is force, m is mass, and a is acceleration.

2. How do you calculate force using Newton's Second Law?

To calculate force using Newton's Second Law, you need to know the mass of the object and its acceleration. Multiply the mass by the acceleration to find the force. For example, if an object has a mass of 5 kg and is accelerating at a rate of 10 m/s^2, the force would be 50 Newtons (N).

3. What is the unit of measurement for force in Newton's Second Law?

The unit of measurement for force in Newton's Second Law is Newtons (N). One Newton is equal to 1 kilogram-meter per second squared (kg•m/s^2).

4. Why is acceleration squared in the equation F=ma?

Acceleration is squared in the equation F=ma because it is a measure of how quickly the velocity of an object changes. This means that acceleration has both a magnitude (speed) and a direction, and squaring it accounts for both of these factors in the calculation of force.

5. How do you calculate the velocity squared (v^2) using Newton's Second Law?

To calculate the velocity squared (v^2) using Newton's Second Law, you need to know the force acting on the object and its mass. Divide the force by the mass to find the acceleration, and then multiply the acceleration by 2. The result will be the velocity squared. For example, if the force acting on an object is 20 N and its mass is 5 kg, the acceleration would be 4 m/s^2. Multiplying this by 2 would give a velocity squared of 8 m^2/s^2.

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