Physics ball thrown in air question

[physics120]

Homework Statement

Five balls move through the air. All five have the same size and shape. Air resistance is not negligible. Rank in order, from largest to smallest, the magnitudes of the accelerations a1 to a5. Some may be equal. Give your answer in the form 1>2=3>4>5 and explain your ranking.

where Vy= velocity in the y-direction

A1: 50 g, just released, Vy=0
A2: 100 g, just released, Vy=0
A3: 50 g, Vy= -20 m/s
A4: 100 g, Vy= -20 m/s
A5: 50 g, Vy= 20 m/s

Homework Equations

1)
where D= drag,
A= cross-section area (which in this question is EQUAL for every ball),
v= velocty

vector D = ((1/4)Av^2, direction opposite the motion)

2)
where (Fnet)y= net force in the y direction,
m= mass

Acceleration going up: a= (Fnet)y/m
a= (-mg - D)/m
a= -(g + (D/m))
The magnitude of acceleration going up, which is the ball's deceleration as it rises, is (g + (D/m)).

As the ball falls, its acceleration going down is:

a= (Fnet)y/m
a= (-mg + D)/m
a= -(g - (D/m))

The magnitude of the acceleration going down is (g - (D/m))

The Attempt at a Solution

5>1=2>4>3



AM I CORRECT? PLEASE HELP. thank-you!

 

[Carid]

I'm not happy with number 5. This ball experiences more drag than ball A1.

 

[physics120]

I'm not happy with number 5. This ball experiences more drag than ball A1.

Ball number 5 experiences more drag, but it moves in the downward direction, along with its gravitational force. So, it experiences a greater net force in the downward direction, which means a greater acceleration than ball 1 since ball one has no drag force and only an accleration of -g from its gravitational force in the downward direction.

That is what I think, anyway. Let me know if you see it a different way!

 

[LowlyPion]

So, it experiences a greater net force in the downward direction, ...

How is that?

Air resistance is ∝ v, but is in the direction opposing v.

Starting from rest it's contribution is negligible. At a greater speed, its retarding effect is upward, however, against the direction of motion.

 

[nvn]

Carid: Yes, ball A5 experiences more air drag than A1. That is why the absolute value (magnitude) of the acceleration of A5 is greater than A1.

LowlyPion: Ball A5 is moving upward, even though physics120 claims it is moving downward in post 3, which contradicts post 1. This appears to be a typographic mistake by physics120 in post 3.

physics120: Nice work. Your final answer in post 1 is correct; 5 > 1 = 2 > 4 > 3. Regarding your relevant equations, I might disagree with your air drag force equation. I thought air drag force is Fd = 0.5*rho*Cd*A*v^2, where Cd = 0.47 for a sphere. If so, this would give Fd = 0.288*A*v^2, instead of 0.25*A*v^2. However, making this minor correction does not change the outcome for your answer in post 1, which is correct either way. Other than my above comment, your work under relevant equations in post 1 looks correct. And, as I noted above, it appears you made a typographic mistake in post 3.

 

[LowlyPion]

Oops. Thanks for catching that. I thought it was 3. That is correct.

 

[physics120]

Carid: Yes, ball A5 experiences more air drag than A1. That is why the absolute value (magnitude) of the acceleration of A5 is greater than A1.

LowlyPion: Ball A5 is moving upward, even though physics120 claims it is moving downward in post 3, which contradicts post 1. This appears to be a typographic mistake by physics120 in post 3.

The way I worded it may have been unclear. When I said that ball number 5 experiences more drag, but it moves in the downward direction, I meant the DRAG not the BALL moves in the downward direction. The ball is moving upward. However, drag goes downward since drag is always in the direction opposite of motion.

physics120: Nice work. Your final answer in post 1 is correct; 5 > 1 = 2 > 4 > 3. Regarding your relevant equations, I might disagree with your air drag force equation. I thought air drag force is Fd = 0.5*rho*Cd*A*v^2, where Cd = 0.47 for a sphere. If so, this would give Fd = 0.288*A*v^2, instead of 0.25*A*v^2. However, making this minor correction does not change the outcome for your answer in post 1, which is correct either way.

I took that equation out of my physics120 second edition textbook. They did say, however, that D "approximately equals" ((1/4)Av^2, direction opposite the motion). I just put "=" instead of "approx =" because in this question, since A is equal in all balls, all we need to use from this equation is the fact that D is proportionally equal to the square of v.


Thank-you for your help!