Relation between the velocity vector and the acceleration vector of an object

AI Thread Summary
In uniform circular motion, the position vector is expressed as u=cos(wt)i +sin(wt)j, with the velocity and acceleration vectors being constant in magnitude and perpendicular at each instant. Knowing the velocity vector at a specific time allows for the calculation of angular frequency (ω), which can then be used to predict the position vector at any future time. This relationship holds true because the motion is circular and uniform, providing a consistent framework for analysis. Additionally, the acceleration vector can also be derived similarly, reinforcing the predictability of the motion. Thus, both vectors provide essential information for determining future positions in circular motion.
Clockclocle
Messages
31
Reaction score
1
Homework Statement
velocity vector and acceleration vector
Relevant Equations
u=cos(wt)i +sin(wt)j
A uniform circular motion of a point always yield an equation u=cos(wt)i +sin(wt)j of position vector. Which we deduce the acceleration and velocity vector with constant magnitude and they are perpendicular at each instant. Can I use the information of them at one instant to predict the position vector at any instant later? Or they are only provide the information that the point is moving in the direction of velocity vector and also is affected by a centripetal force point toward the center of the circle?
 
Physics news on Phys.org
Clockclocle said:
Homework Statement: velocity vector and acceleration vector
Relevant Equations: u=cos(wt)i +sin(wt)j

A uniform circular motion of a point always yield an equation u=cos(wt)i +sin(wt)j of position vector. Which we deduce the acceleration and velocity vector with constant magnitude and they are perpendicular at each instant. Can I use the information of them at one instant to predict the position vector at any instant later? Or they are only provide the information that the point is moving in the direction of velocity vector and also is affected by a centripetal force point toward the center of the circle?

Somewhat. If you already know the motion is circular around the origin with radius 1 and uniform, you can write $$\vec u = \cos(\omega t)\hat \imath+\sin(\omega t)\hat \jmath)\tag{1}$$
for the position vector, as you did.

Taking the derivative ##\vec v \equiv {d\vec u\over dt}## gives $$\vec u = \omega\left (-\sin(\omega t)\hat \imath+\cos(\omega t)\hat \jmath)\right )\tag {2}$$

Knowing the velocity vector at some instant ##t_0## would give you ##\omega## and ##t_0##, which you can use in ##(1)## for any ##t##.

And you could also do something similar from ##\vec a##, the second derivative.

##\ ##
 
Kindly see the attached pdf. My attempt to solve it, is in it. I'm wondering if my solution is right. My idea is this: At any point of time, the ball may be assumed to be at an incline which is at an angle of θ(kindly see both the pics in the pdf file). The value of θ will continuously change and so will the value of friction. I'm not able to figure out, why my solution is wrong, if it is wrong .
Thread 'Trying to understand the logic behind adding vectors with an angle between them'
My initial calculation was to subtract V1 from V2 to show that from the perspective of the second aircraft the first one is -300km/h. So i checked with ChatGPT and it said I cant just subtract them because I have an angle between them. So I dont understand the reasoning of it. Like why should a velocity be dependent on an angle? I was thinking about how it would look like if the planes where parallel to each other, and then how it look like if one is turning away and I dont see it. Since...
Thread 'Voltmeter readings for this circuit with switches'
TL;DR Summary: I would like to know the voltmeter readings on the two resistors separately in the picture in the following cases , When one of the keys is closed When both of them are opened (Knowing that the battery has negligible internal resistance) My thoughts for the first case , one of them must be 12 volt while the other is 0 The second case we'll I think both voltmeter readings should be 12 volt since they are both parallel to the battery and they involve the key within what the...
Back
Top