# Physics ball thrown in air question

• physics120
In summary, the five balls experience air resistance, with ball A5 experiencing the most due to its higher velocity. When comparing the magnitudes of their accelerations, ball 5 has the greatest magnitude of acceleration followed by balls 1 and 2 having equal magnitudes, followed by ball 4 and then ball 3 having the smallest magnitude. This is because ball 5 has the greatest net force in the downward direction due to air resistance and gravity, while ball 3 has the smallest net force in the downward direction. The equations used to determine this ranking were Fd = 0.25Av^2 and a = -(g + (D/m)) for the upward acceleration and a = -(g - (D/m)) for the
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

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

Carid said:
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!

physics120 said:
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.

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.

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

nvn said:
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.

## 1. What is the equation for the motion of a ball thrown in the air?

The equation for the motion of a ball thrown in the air is y = y0 + v0t - 1/2gt^2, where y is the vertical displacement, y0 is the initial height, v0 is the initial velocity, t is the time, and g is the acceleration due to gravity.

## 2. How does the angle at which a ball is thrown affect its trajectory?

The angle at which a ball is thrown affects its trajectory by determining the initial vertical and horizontal components of its velocity. A ball thrown at a higher angle will have a greater initial vertical velocity and therefore reach a greater height, while a ball thrown at a lower angle will have a greater initial horizontal velocity and therefore travel farther horizontally.

## 3. What affects the maximum height and range of a ball thrown in the air?

The maximum height and range of a ball thrown in the air are affected by the initial velocity, angle at which it is thrown, and the acceleration due to gravity. The higher the initial velocity and angle, the greater the maximum height and range. The acceleration due to gravity decreases the height and range of the ball.

## 4. How does air resistance affect the motion of a ball thrown in the air?

Air resistance, also known as drag, opposes the motion of a ball thrown in the air. It affects the ball's trajectory by decreasing its velocity and therefore decreasing its range. Air resistance is more significant for lighter and slower moving objects, so it may have a larger impact on a lighter ball thrown at a lower velocity.

## 5. How does the surface on which a ball is thrown affect its motion?

The surface on which a ball is thrown can affect its motion by providing resistance or friction. A rough surface will cause more friction and decrease the ball's range, while a smooth surface will cause less friction and allow the ball to travel farther. The surface can also affect the ball's bounce after it has landed, with a softer surface causing a lower bounce and a harder surface causing a higher bounce.

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