Question About an F=ma Problem with a small ball impacting a ramp

[mopit_011]

Homework Statement

At time t = 0, a small ball is released on the track shown, with an initial rightward velocity. Assume the ball always rolls along the track without slipping.

Relevant Equations

N/A

(This is the diagram from the problem.)

My professor assigned this as extra practice and I don’t understand why the answer is (C) instead of (B). Why would sharp bends be the correct answer? Isn’t the curved diagram more accurate for movement on a slope?

 

[PeroK]

I'd say it's impossible to choose between B and C as it depends on how you model the transition to the slope.

 

[Drakkith]

Why would sharp bends be the correct answer? Isn’t the curved diagram more accurate for movement on a slope?

The track has a sharp, instant bend in it, so the balls acceleration also has a sharp, instant change. Just before the ball hits the bend it has an acceleration of zero. Right after the bend it has an acceleration of $9.8*sin(\theta)$ where $\theta$ is the angle of the slope.

 

[haruspex]

Homework Statement:: At time t = 0, a small ball is released on the track shown, with an initial rightward velocity. Assume the ball always rolls along the track without slipping.
Relevant Equations:: N/A

(This is the diagram from the problem.)
View attachment 298815

My professor assigned this as extra practice and I don’t understand why the answer is (C) instead of (B). Why would sharp bends be the correct answer? Isn’t the curved diagram more accurate for movement on a slope?

Write the equation for constant acceleration: v=at. v is a linear function of time.
You may be intuitively thinking it should be curved like B because the height will be quadratic with time.
But in my view none of the diagrams is quite right. If the bend is that sharp then there is a sudden impulse, dropping the speed instantly to $v\cos(\theta)$. To make it look like C the transition to the ramp must have a radius of curvature greater than that of the ball.
It will drop again when it returns to ground.

 

[Orodruin]

I'd say it's impossible to choose between B and C as it depends on how you model the transition to the slope.

... or even if the transition to the slope should be considered elastic ...

 

[haruspex]

... or even if the transition to the slope should be considered elastic ...

Merely being elastic doesn't help since that would mean the ball becomes airborne.

 

[Orodruin]

I'd rather say speed conserving. Merely being elastic doesn't help since that would mean the ball becomes airborne.

This was kind of the point. If it is speed conserving in a sharp bend, then the ball bounces off the ramp.

This was likely not considered by the person constructing the problem.

 

[haruspex]

This was kind of the point. If it is speed conserving in a sharp bend, then the ball bounces off the ramp.

This was likely not considered by the person constructing the problem.

Yes, I deleted speed conserving, it wasn't quite what I meant.
If the radius of the bend is less than the radius of the ball then either it will bounce off or it will lose speed in the impact, probably both.

 

[Drakkith]

Obviously the question is a simplified scenario that ignores things like the size and geometry of the ball and track. Spherical cows and all that, you know.

 

[haruspex]

Obviously the question is a simplified scenario that ignores things like the size and geometry of the ball and track. Spherical cows and all that, you know.

It is not really a question of simplification. The diagram clearly shows a ball with a radius greater than in the angle of the slope. We can only infer that the transition to the slope is supposed to be smooth because there is no option that matches an abrupt change. Just add such an option and immediately the correct answer changes. Or if we take away the multiple choice and ask the student to sketch the graph the alert student will show two sudden drops in speed.
Problem setter assessment: must try harder.

 

[nasu]

No matter how you model the transition, if the ball moves along the inclined plane the velocity decreases linearly. So (B) is not good. It shows curved lines not only at the transition but for the whole motion on the incline.

 

[Drakkith]

It is not really a question of simplification.

I disagree. This is obviously a problem for students just beginning their physics education and as such it simplifies things greatly to keep students from being overwhelmed, as all problems at this level do. There's as much reason to bring in the shape and size of the ball in this problem as there is in bringing in air resistance in a problem regarding the acceleration of a box sitting on the back of a braking truck.