# Help Evaluating Mathematical Modelling of Physical Problems

## 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 modelled 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 modelled 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 modelled 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?

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PeroK
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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?

haruspex
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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".
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?

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.

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.

PeroK
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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?

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 modelled 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?

PeroK
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Only because it is modelled 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!

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.

PeroK
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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.

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.

PeroK
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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?

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.

PeroK
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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.

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 modelled? But how does this affect air resistance?

PeroK
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Compare what I said:

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

the acceleration is the same regardless of the object being modelled?
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 ?

PeroK
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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.

PeroK
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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?

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.

PeroK
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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.

PeroK