Velocity and Acceleration Vectors

In summary, the conversation is about finding the vector for velocity using rotational kinematic equations and the position vector. It involves expressing r and theta as functions of time, taking derivatives, and differentiating to find x and y components of velocity.
  • #1
CGI
74
1

Homework Statement


upload_2016-1-23_15-23-7.png


Homework Equations


Rotational Kinematic Equations
Kinematic Equations

The Attempt at a Solution


I honestly have no clue how to get a vector out of this.
I thought about an equation:

Θ = Θ(initial) + ω(initial)*t + .5αt^2

and how maybe that v = wr could play into this, but there is so much I don't know where to start.

Any help would be really appreciated!
 
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  • #3
Oh okay, so I should be thinking of something along those lines.

So if my theta = 0 and my R = .8ft, would my position vector just be r = .8êr?
what would my r dot and theta dot be? Sorry, this is still relatively new to me
and I've been watching videos on it as well. I just want to make sure I understand
this.
 
  • #4
CGI said:
Oh okay, so I should be thinking of something along those lines.

So if my theta = 0 and my R = .8ft, would my position vector just be r = .8êr?
what would my r dot and theta dot be? Sorry, this is still relatively new to me
and I've been watching videos on it as well. I just want to make sure I understand
this.
Assuming the answers are wanted in terms of the x, y coordinates:
Let the diagram position represent time 0. At time t, what will r and theta be? So what will x and y be?
 
  • #5
Would the x just be rcosΘ and y be rsinΘ?
 
  • #6
CGI said:
Would the x just be rcosΘ and y be rsinΘ?
Yes, but you are trying to find velocities and accelerations, so you need to express them as functions of time.
 
  • #7
Oh okay. Right. Could that just be rcostheta(t) and rsintheta(t)?
 
  • #8
CGI said:
Oh okay. Right. Could that just be rcostheta(t) and rsintheta(t)?
According to the problem statement, both r and theta vary with time. Write each as a function of t.
 
  • #9
Hmmm...okay. Could I say that r = r_initial + vt and that theta = theta_inital + wt where w = 45 rev/min?
 
  • #10
CGI said:
Hmmm...okay. Could I say that r = r_initial + vt and that theta = theta_inital + wt where w = 45 rev/min?
Yes. Now express x and y that way and differentiate as necessary.
 
  • #11
When you say express x and y in that way, do you mean that I can say

x = (r_initial + vt)cos(Θ_initial + ωt)

And the same for y, only with a "sin?"
 
  • #12
CGI said:
When you say express x and y in that way, do you mean that I can say

x = (r_initial + vt)cos(Θ_initial + ωt)

And the same for y, only with a "sin?"
Yes. Now differentiate.
 
  • #13
Okay I was just double checking. So when I take the derivative with respect to t I get,

x = vcos(Θ_inital + ωt) - (r_initial + vt)sin(Θ_initial + ωt)*ω

y = vsin(Θ_inital + ωt) + (r_inital + vt)cos(Θ_initial + ωt)*ω

Does this look about right?
 
  • #14
CGI said:
Okay I was just double checking. So when I take the derivative with respect to t I get,

x = vcos(Θ_inital + ωt) - (r_initial + vt)sin(Θ_initial + ωt)*ω

y = vsin(Θ_inital + ωt) + (r_inital + vt)cos(Θ_initial + ωt)*ω

Does this look about right?
Yes.
 
  • #15
Okay great. So would these two x and y be the vector for velocity?
 
  • #16
CGI said:
Okay great. So would these two x and y be the vector for velocity?
Small correction: in your post #13 I presume you meant ##\dot x=## etc., not x=. Similarly y.
If so, yes they would be the x and y components of velocity.
 

FAQ: Velocity and Acceleration Vectors

What is a velocity vector?

A velocity vector is a mathematical representation of the speed and direction of an object's motion. It is typically represented by an arrow, with the length representing the speed and the direction representing the direction of motion.

What is an acceleration vector?

An acceleration vector is a mathematical representation of the rate at which an object's velocity is changing. It is also typically represented by an arrow, with the length and direction representing the magnitude and direction of the acceleration.

How are velocity and acceleration vectors related?

Velocity and acceleration vectors are related in that acceleration is the rate of change of velocity. This means that acceleration is the change in velocity over time, and can be represented by a change in the direction or magnitude of the velocity vector.

How do you calculate the magnitude of a velocity or acceleration vector?

The magnitude of a velocity or acceleration vector can be calculated using the Pythagorean theorem. For a 2-dimensional vector, the magnitude is equal to the square root of the sum of the squares of the vector's components. For a 3-dimensional vector, the magnitude is equal to the square root of the sum of the squares of the vector's x, y, and z components.

Can velocity and acceleration vectors change over time?

Yes, velocity and acceleration vectors can change over time. This is because velocity and acceleration are both dependent on time, and can be affected by external forces such as gravity or friction. Therefore, the magnitude and direction of velocity and acceleration vectors can change as the object's motion changes.

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