Trajectory of a turning particle

[rovim]

In this problem, I need to find the trajectory of a particle (as a function of time) which moves at a speed 's' but also turns at an increasing rate; angular acceleration α. The trajectory looks like a spiral which converges to a point.

The particle has an initial position vector p and a velocity vector v. So without an angular velocity or 'turning effect' the particle should simply trace out the path "p(t) = ∫ v dt", however the particle has an angular acceleration which means that the velocity vector is rotated at the rate α, therefore v_new = R(Δθ) v_old (R(θ) is the rotation matrix to rotate a 2d vector θ degrees).

I'm looking at this problem in a 2d sense.

An example of this would be a car that is traveling a constant speed but the driver starts turning the wheel at a constant rate so that the car turns sharper and sharper.

I've tried tackling this problem by expressing the velocity vector and angular velocity as a complex numbers and integrating their product, as well as trying a method using the rotation matrix, but on both occasions I'm stuck with 'dt's' inside trigonometric functions while integrating.

 

[haruspex]

In this problem, I need to find the trajectory of a particle (as a function of time) which moves at a speed 's' but also turns at an increasing rate; angular acceleration α. The trajectory looks like a spiral which converges to a point.

The particle has an initial position vector p and a velocity vector v. So without an angular velocity or 'turning effect' the particle should simply trace out the path "p(t) = ∫ v dt", however the particle has an angular acceleration which means that the velocity vector is rotated at the rate α, therefore v_new = R(Δθ) v_old (R(θ) is the rotation matrix to rotate a 2d vector θ degrees).

I'm looking at this problem in a 2d sense.

An example of this would be a car that is traveling a constant speed but the driver starts turning the wheel at a constant rate so that the car turns sharper and sharper.

I've tried tackling this problem by expressing the velocity vector and angular velocity as a complex numbers and integrating their product, as well as trying a method using the rotation matrix, but on both occasions I'm stuck with 'dt's' inside trigonometric functions while integrating.

Let $\hat k$ be the unit vector orthogonal to the plane of motion. Can you write a vector equation that relates $\dot {\vec v}$ to $\vec v$, $\hat k$ and some scalar function $\alpha(t)$? Done correctly, it will entail the information that the speed is constant.

 

[rovim]

Let $\hat k$ be the unit vector orthogonal to the plane of motion. Can you write a vector equation that relates $\dot {\vec v}$ to $\vec v$, $\hat k$ and some scalar function $\alpha(t)$? Done correctly, it will entail the information that the speed is constant.

Thanks for responding, but I don't really grasp your idea with my limited knowledge of vector calculus. Wouldn't adding a perpendicular component cause the speed to grow unboundedly? To me, it doesn't seem to be a true rotation. I might be completely off track, so if someone could provide me with a semi-detailed derivation or point to a similar problem, that would be a life saver.

 

[haruspex]

Thanks for responding, but I don't really grasp your idea with my limited knowledge of vector calculus. Wouldn't adding a perpendicular component cause the speed to grow unboundedly? To me, it doesn't seem to be a true rotation. I might be completely off track, so if someone could provide me with a semi-detailed derivation or point to a similar problem, that would be a life saver.

If you want the particle to turn there has to be a perpendicular component. If you want the speed constant, there can only be a perpendicular acceleration. If it is a constant magnitude acceleration the particle will move in a circle, but you wanted a spiral, so it has to vary. (I did not suggest it should increase without limit.)

 

[rovim]

If you want the particle to turn there has to be a perpendicular component. If you want the speed constant, there can only be a perpendicular acceleration. If it is a constant magnitude acceleration the particle will move in a circle, but you wanted a spiral, so it has to vary. (I did not suggest it should increase without limit.)

Ah yes I understand that now. With some working out I end up with the equation:
Δp/Δt = s$\hat v$ + $\frac{1}{2}$ srt²$\hat k$

v is the direction of the velocity, s is the speed of the movement and r is the angular acceleration.

Now, I have absolutely no idea to integrate this to get an expression for p.

 

[haruspex]

Ah yes I understand that now. With some working out I end up with the equation:
Δp/Δt = s$\hat v$ + $\frac{1}{2}$ srt²$\hat k$

v is the direction of the velocity, s is the speed of the movement and r is the angular acceleration.

Now, I have absolutely no idea to integrate this to get an expression for p.

I defined $\hat k$ as orthogonal to the plane of motion. The lateral acceleration is therefore orthogonal to that. Since you want constant speed, we have $|\vec v|^2=\vec v.\vec v$ constant. What do you get if you differentiate that that? Can you then see how to write $\dot {\vec v}$ in terms of $\vec v$ and $\hat k$ ?

 

[rovim]

I defined $\hat k$ as orthogonal to the plane of motion. The lateral acceleration is therefore orthogonal to that. Since you want constant speed, we have $|\vec v|^2=\vec v.\vec v$ constant. What do you get if you differentiate that that? Can you then see how to write $\dot {\vec v}$ in terms of $\vec v$ and $\hat k$ ?

Hmmm. Differentiating $|\vec v|^2=\vec v.\vec v$ only seems to prove the result $\dot {\vec v} .\vec v = 0$ which gives me the idea that I should completely replace the $\hat k$ component with $\dot {\vec v}$. But still I haven't been able find the magnitude.

I believe I'm lacking some essential knowledge in vector calculus to be able to solve this. So putting this as mathematically as I can, the problem describes 4 equations I really don't know how to put together to find p(t):
$\vec p (t) = \int_0^t \! \vec v (t) \, \mathrm{d}t + p_{initial}$
$\vec v (t) = \int_0^t \! \ \dot{\vec v} (t) \, \mathrm{d}t + v_{initial}$
$\dot {\vec v} \cdot \vec v = 0$
$| \dot {\vec v} (t) | = \alpha (t) | \vec v (t) |$

 

[haruspex]

Hmmm. Differentiating $|\vec v|^2=\vec v.\vec v$ only seems to prove the result $\dot {\vec v} .\vec v = 0$ which gives me the idea that I should completely replace the $\hat k$ component with $\dot {\vec v}$. But still I haven't been able find the magnitude.

I believe I'm lacking some essential knowledge in vector calculus to be able to solve this. So putting this as mathematically as I can, the problem describes 4 equations I really don't know how to put together to find p(t):
$\vec p (t) = \int_0^t \! \vec v (t) \, \mathrm{d}t + p_{initial}$
$\vec v (t) = \int_0^t \! \ \dot{\vec v} (t) \, \mathrm{d}t + v_{initial}$
$\dot {\vec v} \cdot \vec v = 0$
$| \dot {\vec v} (t) | = \alpha (t) | \vec v (t) |$

$\dot {\vec v} .\vec v = 0$ tells you that those two vectors are orthogonal. $\dot {\vec v}$ is also orthogonal to $\hat k$. If a vector is known to be orthogonal to two others, how can you write it as a scalar multiple of a certain function of those two vectors?

 

[rovim]

$\dot {\vec v} .\vec v = 0$ tells you that those two vectors are orthogonal. $\dot {\vec v}$ is also orthogonal to $\hat k$. If a vector is known to be orthogonal to two others, how can you write it as a scalar multiple of a certain function of those two vectors?

So $\vec v = \alpha (t) \vec v \times \hat k$?

 

[rovim]

It's been a few days and I still have no idea. Could anyone atleast show me what topics to look into to solve this problem?