Which statement is true? (Velocity and Acceleration of a Tennis Ball)

[erobz]

Sorry @haruspex , I can’t understand your example.
Probably irrelevantly and wrongly, I have just tried to extend post #23 from @jbriggs444 by excluding the action of the unidirectional influence of g, as the problem in discussion gives the constrain of a rotating acceleration vector.

In my humble opinion, problems that may present vectors that change, rotate or move erratically without a natural reason are not helpful to students.
I hope that @paulimerci understands what the correct answer is and the natural or physical reason behind it.

This was just supposed to be a side exploration, @paulimerci is aware of that. However, I probably should have started a new thread.

Don't you think there is also an importance to understanding kinematics in physics? It's the mathematical framework in which we evaluate the physical laws. If you don't understand it well, you could end up doing what I did! What I don't understand is still a mystery to me, but it's clear I don't understand something. I could just as easily fumble "that something" applying the physical laws IMO.

Anyhow, just a thought experiment. I didn't mean to cause strife.

 

[kuruman]

I think this discussion is moving in a direction that is unlikely to help the OP. The original question is not about the trajectory of the tennis ball but about what happens to the speed and the direction of motion when the angle between the velocity and the acceleration is 120°. Whether that angle is instantaneously at that value or, as the problem states, it is maintained "during an interval of time" is irrelevant to which of the four listed choices is correct. If one of the answers is correct at an instant of time, it will be correct at subsequent instants of time as long as the angle is maintained at 120°.

Thus, let's project the ball at 45° relative to the horizontal and ask the question, "which of the four choices is correct when the velocity is at 30° relative to the horizontal?"

 

[erobz]

If anyone could just PM me where my first mistake is (unless it's all crap) I'd appreciate it.

 

[Lnewqban]

If anyone could just PM me where my first mistake is (unless it's all crap) I'd appreciate it.

What makes you believe that you made a mistake?
I see none.

 

[erobz]

What makes you believe that you made a mistake?
I see none.

It's telling me the trajectory is a line? The angle $\theta$ is fixed for a line trajectory ( and not just any line, a very specific $45^{\circ}$).

But the I'm also getting from the ODE:

$$ \frac{d \theta}{dx} = a \sin \beta \cos \theta $$

We get:

$$ \ln \left| \frac{ \sec \theta + \tan \theta }{ \sec \theta_o + \tan \theta_o} \right| = a \sin \beta ( x - x_o ) $$

Is that really yield a constant $\theta(x)$? Seems like a contradiction to me.

Even what @kuruman is saying in #37 could not be the case if that following math in #34 were correct?

 

[Lnewqban]

It's telling me the trajectory is a line? The angle $\theta$ is fixed for a line trajectory ( and not just any line, a very specific $45^{\circ}$).

But the I'm also getting from the ODE:

$$ \frac{d \theta}{dx} = a \sin \beta \cos \theta $$

We get:

$$ \ln \left| \frac{ \sec \theta + \tan \theta }{ \sec \theta_o + \tan \theta_o} \right| = a \sin \beta ( x - x_o ) $$

Is that really yield a constant $\theta(x)$? Seems like a contradiction to me.

Even what @kuruman is saying in #37 could not be the case if that following math in #34 were correct?

The problem shows the angle between the v and a vectors to be constant for a brief period of time, but allows freedom for the magnitudes of both vectors to change during the same time.
Therefore, to me, it seems to be physically possible.
I like @kuruman proposal.

 

[erobz]

The problem shows the angle between the v and a vectors to be constant for a brief period of time, but allows freedom for the magnitudes of both vectors to change during the same time.
Therefore, to me, it seems to be physically possible.
I like @kuruman proposal.

For the OP, they don't specify constant magnitude acceleration, just direction. So yeah, no problems there (that jump out at least).

My analysis is supposed to pertain to a trajectory where the magnitude of the acceleration is fixed, it is in that assumption that it seems like a contradiction to me, or something is not correct in what I've done. I honestly can't tell. By inspection I think it should be a spiral, and the math says its a line...that's a strong disconnect.

 

[Lnewqban]

I see, @erobz
The reason for the spiral is beyond my limited understanding.
Even @kuruman approach in post #37 would be an approximation for the beginning of the parabolic trajectory and for a very short period of time, as the angle between g and v will not stay constant.

 

[jbriggs444]

The reason for the spiral is beyond my limited understanding.

The spiral seems straightforward enough. I assume that we are taking the version of the problem where we have a projectile moving in a plane subject to an acceleration of constant magnitude always angled to the left and back at 120 degrees from the current velocity. An initial guess at the resulting trajectory would be a logarithmic spiral.

Clearly this leads to speed as a linear function of time. The projectile will cease moving at a future moment that can be easily calculated. At this time it will be at the center of the spiral. Velocity will cease to have a direction and the given description of the projectile's motion will cease to be predictive.

We might consider re-casting the problem to consider a projectile moving in the opposite manner -- spiralling outward from the center and using polar coordinates to track its motion.

 

[Lnewqban]

Thank you for that excellent explanation, @jbriggs444

 

[DrClaude]

I was intrigued so I did a numerical simulation. The initial conditions are $\vec{v} (0) = 10 \vec{i}$, $(x,y) = (0,0)$, and the acceleration is taken as $|\vec{a}| = 1$ (all units are arbitrary). The plot shows the $(x,y)$ position at time intervals of 1/10. The simulation ends at $t=20$ when $\vec{v} (0) =0$.

 

[erobz]

$$ \frac{d\theta}{dt} = \frac{a}{v_x \sec^2 \theta } (\sin ( \theta + \beta ) - \cos ( \theta + \beta) \tan \theta ) \tag{7}$$Applying the Chain Rule to (7) to eliminate the parameter $t$:

$$ \frac{d \theta }{dx} = \frac{d\theta}{dt} v_x = \frac{a}{\sec^2 \theta } (\sin ( \theta + \beta ) - \cos ( \theta + \beta ) \tan \theta ) \tag{8}$$

I figured out where I hung myself. It was a subtle mistake.

$$\frac{d\theta}{dt} = \frac{a}{v_x \sec^2 \theta } (\sin ( \theta + \beta ) - \cos ( \theta + \beta) \tan \theta ) \tag{7}$$

Applying the chain rule properly this time:

$$ \frac{d\theta}{dt} =\frac{d \theta}{dx} \frac{dx}{dt} = \frac{d \theta}{dx} v_x \tag{8}$$

Subbing (8) into (7), and reducing trigonometric terms:

$$ \frac{d \theta}{dx} = \frac{a}{v_x^2} \sin \beta \cos \theta \tag{9}$$

I'm sure no one cares, but I don't like to leave it like that.

 

[DrClaude]

I figured out where I hung myself. It was a subtle mistake.

$$\frac{d\theta}{dt} = \frac{a}{v_x \sec^2 \theta } (\sin ( \theta + \beta ) - \cos ( \theta + \beta) \tan \theta ) \tag{7}$$

Applying the chain rule properly this time:

$$ \frac{d\theta}{dt} =\frac{d \theta}{dx} \frac{dx}{dt} = \frac{d \theta}{dx} v_x \tag{8}$$

Subbing (8) into (7), and reducing trigonometric terms:

$$ \frac{d \theta}{dx} = \frac{a}{v_x^2} \sin \beta \cos \theta \tag{9}$$

I'm sure no one cares, but I don't like to leave it like that.

I tried my hand also at an analytical solution, but haven't succeeded yet. However, I am perplexed by your result. $d\theta/dt$ should be independent of $\theta$, by rotational symmetry.

 

[erobz]

I tried my hand also at an analytical solution, but haven't succeeded yet. However, I am perplexed by your result. $d\theta/dt$ should be independent of $\theta$, by rotational symmetry.

I don't know (I'm not smart enough to make that judgement call). Perhaps I bungled it even before I got to that step.

The units work out on that last ODE?

from (9) I need to work out $v_x$ as a function of $\theta$ or $x$. My hope would be to somehow go through:

$$ \frac{dv_x}{dt} = a \cos ( \theta + \beta ) $$

It's nice to have "spiral" confirmation with that simulation though! Thank you!

 

[erobz]

Oh, I think I see a path forward now... by the Chain Rule:

$$ \frac{dv_x}{dt} = v_x\frac{dv_x}{dx} = a \cos ( \theta + \beta) \tag{10} $$

By (9):

$$v_x^2 = a \sin \beta \frac{\cos \theta(x)}{ \theta'(x)} \tag{9'}$$

Differentiate (9') w.r.t $x$ (dropping the function notation):

$$ 2 v_x \frac{dv_x}{dx} =- a \sin \beta \left( \sin \theta + \frac{ \cos \theta}{ \theta'^2} \theta'' \right)$$

Then sub (10) into the result

$$ 2 \cos ( \theta + \beta ) = - \sin \beta \left( \sin \theta + \frac{ \cos \theta}{ \theta'^2} \theta'' \right) \tag{11}$$

Now, we can solve for $\theta(x)$...Theoretically and find $y(x)$ from (1).

However, given the form of (11), I think I'm not going to bother.

 

[paulimerci]

What happens during circular movement?
Could you have another acceleration vector that could be combined with the vector of centripetal acceleration in such a way that the resultant acceleration vector forms a constant angle with the velocity vector of the ball?

I understand that the velocity vector and acceleration vector are perpendicular to each other at every location on the circle in uniform circular motion at a constant speed because the acceleration vector points in the direction of the circle's center and the velocity is always tangent to the circle. Where should I draw "another acceleration vector" that you have pointed out in the above question?

 

[haruspex]

Where should I draw "another acceleration vector" that you have pointed out in the above question?

The centripetal acceleration is the acceleration component normal to the velocity, so any other acceleration must be parallel to it. That's the tangential component.

 

[haruspex]

Oh, I think I see a path forward now... by the Chain Rule:

$$ \frac{dv_x}{dt} = v_x\frac{dv_x}{dx} = a \cos ( \theta + \beta) \tag{10} $$

By (9):

$$v_x^2 = a \sin \beta \frac{\cos \theta(x)}{ \theta'(x)} \tag{9'}$$

Differentiate (9') w.r.t $x$ (dropping the function notation):

$$ 2 v_x \frac{dv_x}{dx} =- a \sin \beta \left( \sin \theta + \frac{ \cos \theta}{ \theta'^2} \theta'' \right)$$

Then sub (10) into the result

$$ 2 \cos ( \theta + \beta ) = - \sin \beta \left( \sin \theta + \frac{ \cos \theta}{ \theta'^2} \theta'' \right) \tag{11}$$

Now, we can solve for $\theta(x)$...Theoretically and find $y(x)$ from (1).

However, given the form of (11), I think I'm not going to bother.

It looks a bit nicer in $(s, \psi)$ coordinates. s is distance along the path, ψ is the angle the path makes to some fixed direction.
$\ddot s=a\cos(\theta)$
$\dot s\dot\psi=a\sin(\theta)$
That's not hard to solve; the hard part is figuring out what it looks like as a curve,

 

[paulimerci]

The centripetal acceleration is the acceleration component normal to the velocity, so any other acceleration must be parallel to it. That's the tangential component.

Does it look like this?

 

[haruspex]

Does it look like this?

Yes, though it does not have to be a circle. The diagram works for any section of the path short enough to be approximated as an arc of a circle.

 

[paulimerci]

Yes, though it does not have to be a circle. The diagram works for any section of the path short enough to be approximated as an arc of a circle.

Thank you, that makes sense!

 

[paulimerci]

Thank you, that makes sense!

Is there anything I missed or need to work on?

 

[Lnewqban]

I understand that the velocity vector and acceleration vector are perpendicular to each other at every location on the circle in uniform circular motion at a constant speed because the acceleration vector points in the direction of the circle's center and the velocity is always tangent to the circle. Where should I draw "another acceleration vector" that you have pointed out in the above question?

You understand correctly.
In accelerated circular movement there is a component of the acceleration that is tangential.

 

[paulimerci]

You understand correctly.
In accelerated circular movement there is a component of the acceleration that is tangential.

View attachment 321158

Oh, thank you so much. A lot is explained by these two diagrams. Worth it. I've learned something far more thoroughly than I anticipated.

 

[erobz]

It looks a bit nicer in $(s, \psi)$ coordinates. s is distance along the path, ψ is the angle the path makes to some fixed direction.

Indeed it does look quite tidy and succinct.

So $\psi$ is my $\theta$, and $\theta$ is my $\beta$. Is that correct?

 

[haruspex]

Indeed it does look quite tidy and succinct.

So $\psi$ is my $\theta$, and $\theta$ is my $\beta$. Is that correct?

Yes.
I don't know what the technical term is for $(s, \psi)$ coordinates. I was introduced to them briefly at high school, but I've not come across them since, and couldn't find a mention on the net.

 

[paulimerci]

You understand correctly.
In accelerated circular movement there is a component of the acceleration that is tangential.

View attachment 321158

Therefore, an object undergoing uniform acceleration is accelerating even though its speed is constant. How does the acceleration vs time graph look?

 

[jbriggs444]

Therefore, an object undergoing uniform acceleration is accelerating even though its speed is constant. How does the acceleration vs time graph look?

You are talking about uniform circular motion, right? So an acceleration versus time graph would involve two dimensions of acceleration versus one dimension of time. It is hard to convey this sort of three dimensional graph on a two dimensional page. However, a quick trip to Google finds this graph with the two dimensions superimposed. Both dimensions of acceleration are simple sine waves -- with a 90 degree offset from each other.

Viewed as vectors rather than as component pairs, the position vector will be moving around in a circular trajectory. The velocity vector will also trace out a circular path in velocity space. The acceleration vector will trace out a circular path in acceleration space. The "jerk" vector will trace out a circular path in "jerk" space. And so on.

Uniform circular motion has an unusual property -- the functions for position, velocity, acceleration, "jerk", "snap", "crackle", "pop" and all further derivatives all have graphs that look exactly the same. Each further derivative is shifted 90 degrees from the previous. For instance, the graph for "snap" will match the graph for position in terms of phase. [The amplitudes may differ, but a careful choice of units can make the amplitudes match as well].

 

[erobz]

Yes.
I don't know what the technical term is for $(s, \psi)$ coordinates. I was introduced to them briefly at high school, but I've not come across them since, and couldn't find a mention on the net.

The first seems pretty obvious as $\dot s$ is tangential to $s$

To derive the second equation, I think what is being said:

$$ \tan ( \Delta \psi ) = \frac{a \sin \theta \Delta t }{ \dot s } $$

such that when we take the limit as $\Delta t \to 0$ we get

$$ \dot s \dot \psi = a \sin \theta $$

Just making sure I understand the justification.

 

[haruspex]

The first seems pretty obvious as $\dot s$ is tangential to $s$

To derive the second equation, I think what is being said:

$$ \tan ( \Delta \psi ) = \frac{a \sin \theta \Delta t }{ \dot s } $$

such that when we take the limit as $\Delta t \to 0$ we get

$$ \dot s \dot \psi = a \sin \theta $$

Just making sure I get the justification.

Yes, that's how I derived it. I also checked it gave the right result for uniform circular motion.

 

[Lnewqban]

Therefore, an object undergoing uniform acceleration is accelerating even though its speed is constant. How does the acceleration vs time graph look?

Sorry, I don’t understand your question.
Could you explain it a little further?

 

[paulimerci]

You are talking about uniform circular motion, right? So an acceleration versus time graph would involve two dimensions of acceleration versus one dimension of time. It is hard to convey this sort of three dimensional graph on a two dimensional page. However, a quick trip to Google finds this graph with the two dimensions superimposed. Both dimensions of acceleration are simple sine waves -- with a 90 degree offset from each other.
View attachment 321203
Viewed as vectors rather than as component pairs, the position vector will be moving around in a circular trajectory. The velocity vector will also trace out a circular path in velocity space. The acceleration vector will trace out a circular path in acceleration space. The "jerk" vector will trace out a circular path in "jerk" space. And so on.

Uniform circular motion has an unusual property -- the functions for position, velocity, acceleration, "jerk", "snap", "crackle", "pop" and all further derivatives all have graphs that look exactly the same. Each further derivative is shifted 90 degrees from the previous. For instance, the graph for "snap" will match the graph for position in terms of phase. [The amplitudes may differ, but a careful choice of units can make the amplitudes match as well].

Thank you!

 

[paulimerci]

Sorry, I don’t understand your question.

jbriggs answered my question. Thank you.

Could you explain it a little further?

 

[kuruman]

For those interested I posted a solution to the trajectory problem here
https://www.physicsforums.com/threa...le-between-acceleration-and-velocity.1049440/