Finding phi constant based on initial position and initial velocity

In summary, to find the value of phi, we can solve for t in the position equation and velocity equation separately, using the given initial position and velocity values. We can then equate the two equations to solve for phi, as we have 2 equations and 1 unknown. Another approach is to solve for t in both equations and then use those values to solve for phi.
  • #1
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[tex]x(t)=Acos(ωt+ϕ)\\ v(t)=-Aωsin(ωt+ϕ)\\ \\ Let\quad n=integer,\quad A=0.5,\quad ω=2.\\ \\ Given\quad initial\quad position\quad (0,\quad 0.25)\quad and\quad initial\quad velocity\quad (0,\quad 1),\quad find\quad ϕ.\\ \\ x(t)=0.5cos(2t+ϕ)\\ 0.25=0.5cos(ϕ)\\ ϕ=±1.0471975512+2πn\\ \\ v(t)=-sin(2t+ϕ)\\ 1=-sin(ϕ)\\ ϕ=-1.5707963268[/tex]

So which one is it? :S

Immediate help is very appreciated!
 
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  • #2


Solve your position equation for t; utilize x=0. Solve your velocity equation for t; you have two components for the velocity, what is its initial magnitude? After that, you can set both equations equal to one another and solve for phi.
 
  • #3


Solving for t yields 4 equations - not 2 (2 equations from position and 2 from velocity).
 
  • #4


You are given (x, y)=(0, 0.25). You have an equation for x. Plug in the value for x, then solve for t. You will have an equation that is a function of phi (1 unknown).

You then have an equation for the velocity given as (vx, vy)=(0, 1). Furthermore, you have an equation for the velocity, which would merely be the magnitude of the force vector. You have two components, and hence can determine the magnitude. You can solve this equation for t, which would also be in terms of phi.

Set these two equations equal to one another and you should be able to solve for phi. 2 equations, 1 unknown.
 
  • #5


The given is in the form of (t, x) and (t, v).
 
  • #6


Okay. What if you ignore the values for t, and solve both of the equations for t. Then use the two to find phi?
 

What is the phi constant?

The phi constant, also known as the golden ratio, is a mathematical constant that has been studied for centuries. It is approximately equal to 1.6180339887, and is often denoted by the Greek letter phi (φ).

How is the phi constant related to initial position and initial velocity?

The phi constant is related to initial position and initial velocity through the equation φ = (1 + √5)/2, which is derived from the golden ratio formula. This formula allows us to calculate the phi constant based on the initial position and initial velocity of an object.

Why is the phi constant important in science?

The phi constant is important in science because it appears in many natural and physical phenomena, such as the arrangement of leaves on a stem, the growth patterns of certain plants, and the structure of some crystals. It also has applications in art, architecture, and design.

How do you find the phi constant based on initial position and initial velocity?

To find the phi constant based on initial position and initial velocity, you can use the formula φ = (1 + √5)/2 and plug in the values for the initial position and initial velocity. Alternatively, you can use a calculator or computer program that has a built-in function for calculating the golden ratio.

Are there any real-life examples of using the phi constant to solve problems?

Yes, there are many real-life examples of using the phi constant to solve problems. As mentioned before, the phi constant is present in many natural and physical phenomena, so it can be used to analyze and understand these phenomena. It can also be used in design, such as in creating aesthetically pleasing compositions or in optimizing the efficiency of structures.

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