# Curved motion, r(φ) when r is accelerating

• kamil.borkowski
In summary, the conversation discusses finding an equation for r(t) to describe the motion of a blue dot in a spring-powered target thrower. Different equations are suggested, including one with an acceleration term, but the specific requirements for r(t) are not fully defined. The conversation also mentions the need to consider forces and the use of a rotating reference frame to calculate r(t). Overall, the main goal is to have an equation for r(t) that includes the angular acceleration of the thrower and can help determine the required force in the spring.
kamil.borkowski
Hi there!

I have a problem to find an equation of r(t), which will help me to describe motion of blue dot. I have seen many cases where there is a linear motion and you can write φ=ωt , r=V0t, but here there is an acceleration. I think I can write φ=εt2/2, but what about r(t) ? I have tried differencial equation
r = at2/2 + r0 → r = (r''-rφ')t2/2 + r0, but not sure if it's right. I need all this to have equation of path r(φ). r0 is initial position of blue dot, where V0=0 and ω=0. Cheers!

Last edited:
kamil.borkowski said:
Hi there!

I have a problem to find an equation of r(t), which will help me to describe motion of blue dot.

There are many possible formulas for r(t) that would produce a curved trajectory. To pose a specific mathematical problem, you have to state a complete set of requirements for r(t). Is this a physics problem? Are forces involved?

Stephen Tashi said:
There are many possible formulas for r(t) that would produce a curved trajectory. To pose a specific mathematical problem, you have to state a complete set of requirements for r(t). Is this a physics problem? Are forces involved?

I am trying to describe a motion of a clay target in a spring-powered target thrower. You can involve some forces (f.ex. friction between target and friction tunnel of an arm), but this complicates case even more. The only constant I can assume is velocity when target leaves the arm (27m/s). Everything else has to be calculated. The main thing is to have ε of this arm, because then from M=I⋅ε we can have required force in a spring.

The instantaneous velocity in polar coordinates is given by the vector $\frac{dr}{dt}r$ in the radial direction and $\frac{d\theta}{dt} \theta$ in the tangential direction. If you want the magnitude of this velocity vector to be 27 m/sec at the time t when the target reaches the end of the arm, you have to keep in mind the contribution of the component $\frac{dr}{dt} r$ to the velocity. It will be important to have a physically reasonable formula for $r(t)$.

kamil.borkowski said:
You can involve some forces (f.ex. friction between target and friction tunnel of an arm), but this complicates case even more.
Without forces, how would you know how the object moves?
kamil.borkowski said:
Everything else has to be calculated.
There are many possible motions that end with a speed of 27m/s.

kamil.borkowski said:
The only constant I can assume is velocity when target leaves the arm (27m/s). Everything else has to be calculated,
I think you at least need to also know the time it needs to get to that speed (from rest?), to get the (constant?) angular acceleration of the thrower. Then I would use the rotating reference frame of the thrower to compute r(t) using the inertial forces there.

## 1. What is curved motion?

Curved motion is any type of motion where an object follows a curved path rather than a straight line. This can occur due to forces acting on the object, such as gravity or acceleration.

## 2. What is r(φ) in relation to curved motion?

r(φ) is a mathematical function that represents the distance from the center of curvature to a point on the curved path at a specific angle (φ). It is used to describe the shape and trajectory of a curved motion.

## 3. How is r(φ) affected by acceleration?

When r is accelerating, the value of r(φ) changes as the object moves along the curved path. This is because the acceleration is constantly changing the object's velocity and direction, causing it to follow a curved path rather than a straight line.

## 4. What is the difference between centripetal and tangential acceleration?

Centripetal acceleration is the acceleration that causes an object to move in a circular path, and is always directed towards the center of the circle. Tangential acceleration is the acceleration that occurs in the direction of motion, and is often caused by a change in speed or direction.

## 5. How is curved motion related to circular motion?

Curved motion is a type of circular motion where an object follows a circular path. However, not all circular motion is curved, as an object can also move at a constant speed in a circular path without changing direction. Curved motion refers specifically to the changing direction and velocity of an object's motion along a curved path.

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