e) What if the same ball were thrown, on Earth, with a higher initial speed? In particular, let's say the initial speed were doubled. What would happen to the maximum height-would it be the same, half the initial value, 3 times the initial value, or what? Explain.
e) What if the same ball were thrown, on Earth, with a higher initial speed? In particular, let's say the initial speed were doubled. What would happen to the maximum height-would it be the same, half the initial value, 3 times the initial value, or what? Explain.
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Part E only please
![3. This is a problem illustrating the use of dimensional analysis. Just by thinking about the dimensionality of
relevant parameters, we can work out the form of a plausible equation to tell us something useful-this is what
was shown in the E = mc² example in lecture. For this problem, we will consider an everyday situation: if you
throw a ball straight up, how high will it go before it starts to fall back down?
This is a question that can be answered precisely using kinematics (as we will see in chapter 2), or using
conservation of energy (as we will see in chapter 7). If you have studied physics before, you may already know
how to use one or both of these methods to derive an equation for the ball's maximum height. It would also
be easy to look up such an equation on the internet (please don't!). But the point of this problem is to see how
much we can figure out just by using dimensional analysis, without using any additional physics knowledge.
We're looking for a height-let's call it h. The height should have dimensions of length: [h] = L. Now, what
might this height depend on? We know from experience that the faster the ball's initial speed, the higher it will
be able to go. So the initial speed, to, will be important. What makes the ball slow down and eventually come
back down? Air resistance might play a part, but let's ignore that for now to keep things as simple as possible.
(Later, we will discuss how to take air resistance into account, and how to tell when it's a good approximation to
neglect it.) Ignoring air resistance, the only thing pulling down on the ball is the Earth's gravity. So we should
expect the Earth's gravitational field strength, g = 9.8 m/s², to be relevant. Also, objects with larger masses feel
more gravitational force (i.e., they are heavier), so the ball's mass, m, may be relevant.
Is there anything else that the maximum height might depend on? There are lots of properties of the ball
that are pretty clearly irrelevant-its color, how much it cost, and so on. The ball's diameter and surface texture
would be important for its interactions with the air, but since we are neglecting air resistance, these will not
play a role. The initial height obviously matters: the higher the ball starts off, the higher it will end up. But let's
agree to zero out the initial height; I will represent the difference between the lowest and highest positions¹.
So we're expecting that I will be determined by some combination of to, 9, and m. We know that these three
quantities cannot be added together, because they have different dimensions. The only thing we can do is
multiply them. We can write
h=C(vo) (m)" (g)".
Here C represents a dimensionless constant (i.e., a pure number with units), and x, y, and represent
powers that o, m, and g might be raised to. The powers x, y, and are also dimensionless; they may be
positive, negative, or zero, and in general they may be integers or fractions.
a) Figure out the appropriate values for the powers x, y, and so that the dimensions match on both sides of
this equation. (Hints: m has dimensions of mass (M); v is a speed, with dimensions L/T; g is an acceleration,
with dimensions L/T². Review the lecture video with the E = mc² dimensional analysis example for help.)
¹The initial height also very slightly affects the value of g-see chapter 13-but we can safely neglect this for now.
b) Explain why it is not possible to determine the value of the constant C using dimensional analysis alone.
Even without knowing the value of C, we can get a lot of information what we've figured out so far about the
equation for h. Answer the following questions based on your results from part (a):
c) What if the mass of the ball were increased (with its initial speed remaining unchanged)? Would the
maximum height of the heavier ball be greater than, less than, or equal to the original? Explain.](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F90ae1aab-f107-47ea-b140-00f7b3a4760d%2Fe72fc86e-aa77-40bb-adfc-6054490d18af%2Fa6v3zpf_processed.jpeg&w=3840&q=75)
Transcribed Image Text:3. This is a problem illustrating the use of dimensional analysis. Just by thinking about the dimensionality of
relevant parameters, we can work out the form of a plausible equation to tell us something useful-this is what
was shown in the E = mc² example in lecture. For this problem, we will consider an everyday situation: if you
throw a ball straight up, how high will it go before it starts to fall back down?
This is a question that can be answered precisely using kinematics (as we will see in chapter 2), or using
conservation of energy (as we will see in chapter 7). If you have studied physics before, you may already know
how to use one or both of these methods to derive an equation for the ball's maximum height. It would also
be easy to look up such an equation on the internet (please don't!). But the point of this problem is to see how
much we can figure out just by using dimensional analysis, without using any additional physics knowledge.
We're looking for a height-let's call it h. The height should have dimensions of length: [h] = L. Now, what
might this height depend on? We know from experience that the faster the ball's initial speed, the higher it will
be able to go. So the initial speed, to, will be important. What makes the ball slow down and eventually come
back down? Air resistance might play a part, but let's ignore that for now to keep things as simple as possible.
(Later, we will discuss how to take air resistance into account, and how to tell when it's a good approximation to
neglect it.) Ignoring air resistance, the only thing pulling down on the ball is the Earth's gravity. So we should
expect the Earth's gravitational field strength, g = 9.8 m/s², to be relevant. Also, objects with larger masses feel
more gravitational force (i.e., they are heavier), so the ball's mass, m, may be relevant.
Is there anything else that the maximum height might depend on? There are lots of properties of the ball
that are pretty clearly irrelevant-its color, how much it cost, and so on. The ball's diameter and surface texture
would be important for its interactions with the air, but since we are neglecting air resistance, these will not
play a role. The initial height obviously matters: the higher the ball starts off, the higher it will end up. But let's
agree to zero out the initial height; I will represent the difference between the lowest and highest positions¹.
So we're expecting that I will be determined by some combination of to, 9, and m. We know that these three
quantities cannot be added together, because they have different dimensions. The only thing we can do is
multiply them. We can write
h=C(vo) (m)" (g)".
Here C represents a dimensionless constant (i.e., a pure number with units), and x, y, and represent
powers that o, m, and g might be raised to. The powers x, y, and are also dimensionless; they may be
positive, negative, or zero, and in general they may be integers or fractions.
a) Figure out the appropriate values for the powers x, y, and so that the dimensions match on both sides of
this equation. (Hints: m has dimensions of mass (M); v is a speed, with dimensions L/T; g is an acceleration,
with dimensions L/T². Review the lecture video with the E = mc² dimensional analysis example for help.)
¹The initial height also very slightly affects the value of g-see chapter 13-but we can safely neglect this for now.
b) Explain why it is not possible to determine the value of the constant C using dimensional analysis alone.
Even without knowing the value of C, we can get a lot of information what we've figured out so far about the
equation for h. Answer the following questions based on your results from part (a):
c) What if the mass of the ball were increased (with its initial speed remaining unchanged)? Would the
maximum height of the heavier ball be greater than, less than, or equal to the original? Explain.
Transcribed Image Text:1The initial height also very slightly affects the value of g-see chapter 13-but we can safely neglect this for now.
b) Explain why it is not possible to determine the value of the constant C using dimensional analysis alone.
Even without knowing the value of C, we can get a lot of information what we've figured out so far about the
equation for h. Answer the following questions based on your results from part (a):
c) What if the mass of the ball were increased (with its initial speed remaining unchanged)? Would the
maximum height of the heavier ball be greater than, less than, or equal to the original? Explain.
d) What if the same ball were thrown with the same initial speed on another planet with stronger gravity.
(i.e., if the value of g were larger?) Would the maximum height of the ball on the new planet be greater than,
less than, or equal to the original? Explain.
e) What if the same ball were thrown, on Earth, with a higher initial speed? In particular, let's say the initial
speed were doubled. What would happen to the maximum height-would it be the same, half the initial value,
3 times the initial value, or what? Explain.
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