# Help Evaluating Mathematical Modelling of Physical Problems

• lpettigrew
In summary, the conversation discusses three different mathematical models for various physical situations. The first model is for a ball being thrown vertically upwards, where it is modeled as a particle moving under constant gravity with no air resistance. The second model is for a puck being hit towards a goal in ice hockey, where the puck is modeled as a particle moving on a smooth surface. The third model is for a cart being pulled by two horses, where all objects are assumed to be particles and there is no resistance in the moving parts of the cart. The conversation also touches on the concept of an inextensible string and the importance of identifying weak points in a model to suggest improvements.
lpettigrew
Homework Statement
Hello, in each of the following a physical problem is given alongside a mathematical model. I have been tasked with deciding whether the model is a good one and suggesting possible improvements.

1) A ball is thrown vertically upwards. Find the time before it gets to its greatest height.
Mathematical model; The tennis ball is modelled as a particle moving under constant gravity with no air resistance.

2. In a game of ice hockey, the puck is hit towards the goal. Find the time it takes to reach the goalkeeper.
Mathematical model; The puck is modelled as a particle moving on a smooth surface.

3. A cart is pulled forward by two horses. Find the tensions in the ropes assuming it is moving with constant speed.

Mathematical Model; The horses and cart are assumed to be particles, the string is assumed to be inextensible and horizontal and there is no resistance in the moving parts of the cart.
Relevant Equations
1) A ball is thrown vertically upwards. Find the time before it gets to its greatest height.
2. In a game of ice hockey, the puck is hit towards the goal. Find the time it takes to reach the goalkeeper.
3. A cart is pulled forward by two horses. Find the tensions in the ropes assuming it is moving with constant speed.
I am really struggling with these problems and do not really know where to begin. In most cases I have just defined what the modelling means, e.g A rough surface is one upon which friction acts etc.
I would really appreciate some help with suitably answering these questions comprehensively, and would note that the textbook I found them in states that each is worth 4 marks, yet I certainly have even unable to make four solid points.
Thank you to anyone who takes the time to offer help and guidance

1) A ball is thrown vertically upwards. Find the time before it gets to its greatest height.

Mathematical model; The tennis ball is modeled as a particle moving under constant gravity with no air resistance.

I know that when an object moves freely under gravity, one can ignore air resistance, which also means that acceleration is constant.
However, the tennis ball in not freely falling since it is accelerating upwards, nor is acceleration constant, so it cannot be assumed that air resistance is negligible.
However, since the ball is modeled as a particle this a fitting analogy and means that when modelling something as a particle, the mass of the object acts through a single point i.e. the size of the object is irrelevant. The particle has no dimensions which means that resistive forces due to e.g. rotation or air resistance can be ignored.2. In a game of ice hockey, the puck is hit towards the goal. Find the time it takes to reach the goalkeeper.

Mathematical model; The puck is modeled as a particle moving on a smooth surface.

Yes, I believe that this is a good model since it would be safe to assume that the puck experiences no friction on the ice, i.e. we could treat the ice as smooth in this case. A smooth surface is frictionless.

3. A cart is pulled forward by two horses. Find the tensions in the ropes assuming it is moving with constant speed.

Mathematical Model; The horses and cart are assumed to be particles, the string is assumed to be inextensible and horizontal and there is no resistance in the moving parts of the cart.

An inextensible string is a string which has a fixed length, it is impossible to stretch, so this statement is appropriate here. Moreover, the depiction of the horses and cart as particles means that mass of the objects acts through a single point and thus the particles do not have resistive forces due to e.g. air resistance. This agrees with the statement that there is no resistance in the moving parts of the cart.

Would an improvement be to assume that the string in light, i.e. it has no mass so no weight acts on a light body?

You are perhaps missing the point. I suggest the first thing is to decide whether the model is "good", "bad" or "fair" by identifying its weak points, if any. Then, by focusing on these you could suggest improvements.

Why not just focus on the tennis ball first. Good, bad or fair?

lpettigrew said:
1) A ball is thrown vertically upwards. Find the time before it gets to its greatest height.

the tennis ball in not freely falling since it is accelerating upwards
I would read it as "a ball has just been thrown upwards".
lpettigrew said:
3. A cart is pulled forward by two horses. Find the tensions in the ropes assuming it is moving with constant speed.
Any diagram? E,g, are these horses parallel or in tandem, and how are the ropes connected?

PeroK said:
You are perhaps missing the point. I suggest the first thing is to decide whether the model is "good", "bad" or "fair" by identifying its weak points, if any. Then, by focusing on these you could suggest improvements.

Why not just focus on the tennis ball first. Good, bad or fair?
Thank you for your reply. I think that this is a good model, since as a particle it's weight acts through a single point resistive forces can be ignored. This is rather true since a ball does not have significant vertices which would otherwise affect such forces. How can I elaborate or offer a more concise and to the point answer though? My apologies I am just a little confused.

haruspex said:
I would read it as "a ball has just been thrown upwards".

Any diagram? E,g, are these horses parallel or in tandem, and how are the ropes connected?
Thank you for your reply, no there are no diagrams included just the physical situation and corresponding model. It is stated that the ropes are inextensible and horizontal.

lpettigrew said:
Thank you for your reply. I think that this is a good model, since as a particle it's weight acts through a single point resistive forces can be ignored. This is rather true since a ball does not have significant vertices which would otherwise affect such forces. How can I elaborate or offer a more concise and to the point answer though? My apologies I am just a little confused.
I would say for a tennis ball it is perhaps a fair model. Why do you say resistive forces can be ignored?

What if we used a table tennis ball? A cricket ball? A golf ball? Would the model be better or worse suited to those?

What about the maximum height itself? What can you say about the relevance of that? What if the maximum height is about ##1m, 10m, 100m##? Does that make any difference?

PeroK said:
I would say for a tennis ball it is perhaps a fair model. Why do you say resistive forces can be ignored?

What if we used a table tennis ball? A cricket ball? A golf ball? Would the model be better or worse suited to those?
Only because it is modeled as a particle. I do not see that the model would be more or less suited to balls of differing size, how does the size of the ball in question affect the suitability of the model here?

lpettigrew said:
Only because it is modeled as a particle. I do not see that the model would be more or less suited to balls of differing size, how does the size of the ball in question affect the suitability of the model here?
That's the question!

PeroK said:
That's the question!

Well, the time taken to reach a larger maximum height would be greater at say 100m than 1m, and if the ball in question was larger I know that heavier objects fall faster, so this again would affect the time to reach its greatest height.

lpettigrew said:
Well, the time taken to reach a larger maximum height would be greater at say 100m than 1m, and if the ball in question was larger I know that heavier objects fall faster, so this again would affect the time to reach its greatest height.
A prerequisite to answering the question is understanding the underlying physics. First, if we ignore air resistance, all objects fall with the same acceleration. "Heavier objects fall faster" is not right at all. A man with an open parachute weighs as much as a man with a packed parachute, but they fall at different rates.

PeroK said:
A prerequisite to answering the question is understanding the underlying physics. First, if we ignore air resistance, all objects fall with the same acceleration. "Heavier objects fall faster" is not right at all. A man with an open parachute weighs as much as a man with a packed parachute, but they fall at different rates.
Right, so in this instance should we ignore air resistance? I do understand that a man with an open parachute weighs as much as a man with a packed parachute, but the open parachute falls at a slower rate since there is a drag force and air resitance acting on the larger surface area.

lpettigrew said:
Right, so in this instance should we ignore air resistance? I do understand that a man with an open parachute weighs as much as a man with a packed parachute, but the open parachute falls at a slower rate since there is a drag force and air resitance acting on the larger surface area.
So, ignoring air resistance for a falling man might be a good model, but it's a bad model for a parachute jump!

That's the sort of question you are trying to answer here. Is ignoring air resistance for a tennis ball a good model?

If you dropped a tennis ball from an aeroplane would it still be valid?

PeroK said:
So, ignoring air resistance for a falling man might be a good model, but it's a bad model for a parachute jump!

That's the sort of question you are trying to answer here. Is ignoring air resistance for a tennis ball a good model?

If you dropped a tennis ball from an aeroplane would it still be valid?
Ignoring air resistance is not a good model, since a different ball would travel upwards with a different acceleration, ie. acceleration is not constant. If the tennis ball was dropped from an aeroplane this model would be better suited because then acceleration would be constant once the ball reched its terminal velocity.

lpettigrew said:
Ignoring air resistance is not a good model, since a different ball would travel upwards with a different acceleration, ie. acceleration is not constant. If the tennis ball was dropped from an aeroplane this model would be better suited because then acceleration would be constant once the ball reched its terminal velocity.
You're probably misunderstanding some of the basics too much. The acceleration due to gravity is constant for all objects (near the Earth's surface). It's the same acceleration on the way up and on the way down.

You need to straighten out those misconceptions first.

PeroK said:
You're probably misunderstanding some of the basics too much. The acceleration due to gravity is constant for all objects (near the Earth's surface). It's the same acceleration on the way up and on the way down.

You need to straighten out those misconceptions first.
Oh, so then this model is suitable because the acceleration is the same regardless of the object being modeled? But how does this affect air resistance?

Compare what I said:

PeroK said:
The acceleration due to gravity is constant for all objects (near the Earth's surface).

with what you said:

lpettigrew said:
the acceleration is the same regardless of the object being modeled?

Do you see the difference?

Yes, you state that the acceleration is due to gravity, which I do not and you state that this is constant for all objects.
So would the suitability of ignoring air resistance depend on the size of the object, ie. the larger the surface area air resistance would have a greater affect like for a parachute. So air resistance is greater for a basketball than a tennis ball ?

lpettigrew said:
So air resistance is greater for a basketball than a tennis ball ?

But so is the force of gravity. Don't confuse "force" with "acceleration".

The problem is that on here we can't just give you the answers and I'm not sure you understand enough physics to answer this question. To analyse whether a mathematical model is good or bad you have to understand the physics pretty well.

Why not google for "when can air resistance be ignored". I found this, for example, which might get you started:

Thank you for the link, from what I gather from the article the air resistance depends on the the density of air and the velocity of the object, not the mass. The result of air resistance on motion is greater as the mass of the object is smaller. So, returning to question 1;
I think that this is a good model as a tennis ball is small and I would assume that the air resistance is likely to be modest also. Ignoring the effect of air resistance would mean that the only force acting on the ball would be its weight. Stating the acceleration due to gravity is constant ignores the variation of gravity with position its on Earth's surface and it's height above sea level. The variation of gravity would not be great for the distance involved either, however, the model may be refined to account for air resistance and the depndence of gravity on height.

lpettigrew said:
Thank you for the link, from what I gather from the article the air resistance depends on the the density of air and the velocity of the object, not the mass. The result of air resistance on motion is greater as the mass of the object is smaller. So, returning to question 1;
I think that this is a good model as a tennis ball is small and I would assume that the air resistance is likely to be modest also. Ignoring the effect of air resistance would mean that the only force acting on the ball would be its weight. Stating the acceleration due to gravity is constant ignores the variation of gravity with position its on Earth's surface and it's height above sea level. The variation of gravity would not be great for the distance involved either, however, the model may be refined to account for air resistance and the depndence of gravity on height.

That looks a lot better. If you had to choose one factor to improve the model would it be a) include air resistance; or, b) include variations in gravity with height?

PeroK said:
That looks a lot better. If you had to choose one factor to improve the model would it be a) include air resistance; or, b) include variations in gravity with height?
I think to include air resistance because I believe that this would have a greater overall effect, since the variations in gravity with height would be very small.

PeroK
Also, I was thinking about question 2. I think that this is also a good model as the puck is small and the ice would offer little resistive force to motion. Although, the model may be improved by accounting for friction and air resistance.

lpettigrew said:
Also, I was thinking about question 2. I think that this is also a good model as the puck is small and the ice would offer little resistive force to motion. Although, the model may be improved by accounting for friction and air resistance.
Yes, that one is simpler.

How could I/ do I need to elaborate further on question 2? Also was it correct to include air resitance as the factor to improve the model, as this would have a greater overall effect, especially as the result of air resistance on motion is greater as the mass of the object is smaller (e.g. a tennis ball) and since the variations in gravity with height would be very small.

lpettigrew said:
How could I/ do I need to elaborate further on question 2? Also was it correct to include air resitance as the factor to improve the model, as this would have a greater overall effect, especially as the result of air resistance on motion is greater as the mass of the object is smaller (e.g. a tennis ball) and since the variations in gravity with height would be very small.
The effect of varying gravity is negligible until the heights involved are significant compared the the radius of the Earth.

Air resistance for a tennis ball has got to be the most significant factor beyond the basic model.

I must admit I have no idea whether air resistance or friction would be more significant for a puck on ice. Something for you to research if you wanted to.

Thank you for setting my down that path, I will do some research and get back to you in a moment whether air resistance or friction is more significant.
I am just evaluating question 3 and then will reply to you with my thoughts...

Thank you very much for all of your help I really appreciate it

PeroK
Question 3)

This is a reasonable model as the extension of the rope is likely to be small so the assumption that the string is inextensible is fitting. Moreover, the statement that there is no resistance in the moving parts of the cart is reasonable if the cart is in optimal working order and is adequately maintained. Although, the model could be refined to take into account that the string may extend. Additionally, it may be preferable to account for the friction and other forces acting due to the moving parts of the cart at a later stage. Moreover, the string may be considered light (meaning that since its mass is very small compared to the other masses in question we can take it to be zero) since weight if the horses and cart far exceeds that of the string. It would also be worthy to take into account the mass of the horses and cart.

lpettigrew said:
Question 3)

This is a reasonable model as the extension of the rope is likely to be small so the assumption that the string is inextensible is fitting. Moreover, the statement that there is no resistance in the moving parts of the cart is reasonable if the cart is in optimal working order and is adequately maintained. Although, the model could be refined to take into account that the string may extend. Additionally, it may be preferable to account for the friction and other forces acting due to the moving parts of the cart at a later stage. Moreover, the string may be considered light (meaning that since its mass is very small compared to the other masses in question we can take it to be zero) since weight if the horses and cart far exceeds that of the string. It would also be worthy to take into account the mass of the horses and cart.

Using this model, what would the tension be in the ropes? Is that realistic?

Would the tension be T=mg+ma? Or would it be that the tension is negligible?

lpettigrew said:
Would the tension be T=mg+ma? Or would it be that the tension is negligible?
The question says constant speed. And nothing about going up hill.

The question does state that it is horizontal, maybe this means that it remains on a level surface also? Using F=ma would this mean that the tension is proportional to the acceleration, it is also constant? If acceleration is proportional to the force F as well as the tension increased acceleration would be contribute to increased force F, which causes proportional increase of tension T. Therefore, the higher acceleration - the higher the tension. When traveling downhill, objects will accelerate wheras when moving uphill they will decelerate and on a flat surface, assuming that there is little friction, they will then maintain a constant speed. So this model is not suitable in that it should not assume that accleration is constant if there is a change in altitude, as this would disrupt the proportional value of the tension?

That's an intetesting take. If I see constant speed in a question I assume acceleration is zero.

Where do these forces come from?

PeroK said:
That's an intetesting take. If I see constant speed in a question I assume acceleration is zero.

Where do these forces cone from?
Sorry, I made a mistake, I mistook the acceleration as constant not the speed. I have puzzled myself.

PeroK said:
That's an intetesting take. If I see constant speed in a question I assume acceleration is zero.

Where do these forces come from?
If the speed is constant then the acceleration would be zero. In accordance with F=ma, if a is zero, then F will zero and since the tension is proportional to the acceleration, would it is also be zero?

lpettigrew said:
If the speed is constant then the acceleration would be zero. In accordance with F=ma, if a is zero, then F will zero and since the tension is proportional to the acceleration, would it is also be zero?
Yes.

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