Two balls with the same air resistance!!!

[peacefulheart]

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1. Two balls with exactly the same size and shape are laughed with same initial velocity from the surface of a perfectly flat plane. When air resistance is considered, the ball with greater mass will have a:

A. longer flight time and a greater maximum height
B. longer flight time and a lower maximum height
C.shorter flight time and a greater maximum height
D. shorter flight time and a lower maximum height.

The answer is A.
The following is how I reasoned:
1.Air resistance affect up trip differently from down trip.

2. For the up trip, Mg+f=Ma1
ma+f=ma2
(f stands for air resistance, which is same for both objects. M is the bigger mass and m is smaller mass., a1 stands for the acceleration of object with greater mass while a2 stands for the acceleration of object with smaller mass.)

a1= g+f/M,
a2=g+f/m
M>m
So, we have a1 is less than a2

Vt=Vo-at, Vt=0,
Vo=at
Vo=a1t1
Vo=a2t2

The two balls have the same initial velocity, since a1<a2, t1>t2.
So, for the up trip, the object with greater mass has longer time

For the height, Vt^2=V0^2-2ah, Vt=o,
V0^2=2ah,
The two balls have the same initial velocity, since a1<a2, h1>h2,
So, the bigger object has a greater maximum height.

3. For the down trip, it is free falling, we all know the time for the bigger object has less time.
Mg-f=Ma1
mg-f=ma2,
a1=g-f/M
a2=g-f/m
M>m
a1>a2,
So, Vt=at if the two balls have the same final velocity, then t1< t2.

4. Combing the results of 2 and 3, we can say that the flight time of the bigger object is greater then smaller one only when air resistance affect up trip much more then down trip.

5. Thanks for your patience to read this long post. Please let me know what you think


thanks a lot.

 

[milski]

All of your logic is generally correct, except you cannot directly determine anything about the times on the way down - the heavier ball is accelerating faster, but it went higher and has longer distance to cover until it falls down. If you want the precise answer, you well have to do it even more detailed that what you've done so far. Which might be interesting for a physics class but is not what the MCAT is about. You will not have time to go in that much detail during the exam.

Here is the simplified version:

For velocities significantly lower than the terminal velocity of the ball, you can consider the up and down motion to be symmetric, so it's only interesting which one will go higher. If there was no air resistance they'd go to the same height. Air resistance affects more the lighter ball, since its larger fraction of the gravity acting on it => the light ball will be more adversely affected and will not go as high as the heavy one.

 

[peacefulheart]

All of your logic is generally correct, except you cannot directly determine anything about the times on the way down - the heavier ball is accelerating faster, but it went higher and has longer distance to cover until it falls down. If you want the precise answer, you well have to do it even more detailed that what you've done so far. Which might be interesting for a physics class but is not what the MCAT is about. You will not have time to go in that much detail during the exam.

Here is the simplified version:

For velocities significantly lower than the terminal velocity of the ball, you can consider the up and down motion to be symmetric, so it's only interesting which one will go higher. If there was no air resistance they'd go to the same height. Air resistance affects more the lighter ball, since its larger fraction of the gravity acting on it => the light ball will be more adversely affected and will not go as high as the heavy one.

1. Thanks for the reply.

2. The reason why I do detailed logical calculation is that it helps me understand concept better.

thanks

 

[milski]

1. Thanks for the reply.

2. The reason why I do detailed logical calculation is that it helps me understand concept better.

thanks

No problem with that - understanding better is always a good thing. It's very important to be able to scale the problem to the level of complexity that's expected when you're tested.

Here is an old thread on a variation of this question - sliding on a ramp with friction. As you can see, if you want to be precise, it gets really complicated, really fast.

http://forums.studentdoctor.net/showthread.php?t=903483

 

[silverice]

All of your logic is generally correct, except you cannot directly determine anything about the times on the way down - the heavier ball is accelerating faster, but it went higher and has longer distance to cover until it falls down. If you want the precise answer, you well have to do it even more detailed that what you've done so far. Which might be interesting for a physics class but is not what the MCAT is about. You will not have time to go in that much detail during the exam.

Here is the simplified version:

For velocities significantly lower than the terminal velocity of the ball, you can consider the up and down motion to be symmetric, so it's only interesting which one will go higher. If there was no air resistance they'd go to the same height. Air resistance affects more the lighter ball, since its larger fraction of the gravity acting on it => the light ball will be more adversely affected and will not go as high as the heavy one.

What about time? It doesn't get slowed down by air resistance as much so it can travel longer?
I just feel like I don't understand the concept welll enough and yet I don't know how can I understand it better.

 

[milski]

What about time? It doesn't get slowed down by air resistance as much so it can travel longer?
I just feel like I don't understand the concept welll enough and yet I don't know how can I understand it better.

For speeds well below terminal velocity, the heavier ball is "less affected" by the air resistance, it will take more time to slow down to zero vertical speed and start falling down. Both time and height for it will have larger values.

I don't like the question that much - it is ambiguous if you understand air resistance really well. To answer it as expected, you have to make a lot of simplifying assumptions.

 

[silverice]

For speeds well below terminal velocity, the heavier ball is "less affected" by the air resistance, it will take more time to slow down to zero vertical speed and start falling down. Both time and height for it will have larger values.

I don't like the question that much - it is ambiguous if you understand air resistance really well. To answer it as expected, you have to make a lot of simplifying assumptions.

Thank you very much