# Velocity of a football with and without drag

• N8G
In summary, the thrower projects a football straight up in the air. There is no air drag on the football, so the speed of the football as a function of height as the ball goes up is a function of the gravitational force and the drag term. Assuming the air drag on the football varies linearly with speed, the speed of the football as a function of height as the ball goes up is a function of the gravitational force and the drag term. Assuming the air drag on the football varies quadratically with speed, the speed of the football as a function of height as the ball goes up is a function of the gravitational force, the drag term, and the square of the speed.
N8G

## Homework Statement

A professional thrower projects a football straight up in the air.
1. Assuming there is no air drag on the football, find the speed of the football as a function of height as the ball goes up.
2. Assuming the air drag on the football varies linearly with speed, find the speed of the football as a function of height as the ball goes up.
3. Assuming the air drag on the football varies quadratically with speed, find the speed of the football as a function of height as the ball goes up.

## Homework Equations

F=ma which extends to mv dv/dx through chain rule
linear drag = c1v
quadratic drag = c2v^2

In each case the sum of the forces in the y direction equals mg minus the corresponding drag term

## The Attempt at a Solution

1. mg = mv dv/dx ... separate variables, v(y) = root( 2gy +vo^2)
2. mg - c1v = mv dv/dx ... unsure how to isolate variables, haven't made it to part 3 but same issue

I'm hoping that I'm missing something simple that I've just overlooked, any help would be greatly appreciated.

Are you confusing x and y, or using them for the same thing?
If up is the positive x (or y) direction, the gravitational force is -mg, not mg.

N8G said:
mg - c1v = mv dv/dx ... unsure how to isolate variables
Try a bit harder... it really is very simple. You just want all the references to x on one side and all the references to v on the other.
And as mjc123 mentions, you should check your signs. Or maybe you are taking g to have a negative value (which is a valid approach).

My bad, any x's should be y's.

For the second part of the problem I have:
mvdv/dy = mg - cv which I reduce to
dv/dy = g/v - c/m
From here I don't see a way to isolate the v term on the rhs from the dy when separating my variables.

And I am taking g to be -9.8m/s^2

Never mind. I think I lost my mind and forgot about how division works.

I should be able to just say:

mg - cv = mv dv/dy

1 = mv/(mg-cv) dv/dy

dy = mv/(mg-cv) dv

Sorry for that.

N8G said:
Never mind. I think I lost my mind and forgot about how division works.

I should be able to just say:

mg - cv = mv dv/dy

1 = mv/(mg-cv) dv/dy

dy = mv/(mg-cv) dv

Sorry for that.
Glad to see you found your mind.

That being said, I figured out how to separate the variables but the integrals for part 2 and 3 both turned out to be horrendous given that I’m looking for the velocity wrt height functions. Each integral needed either aggressive attempts at u substitution or partial fraction decomposition followed by an annoying transform. I find it hard to believe thay professor intended that much work for a minute 10 pt homework assignment. Am I missing something elementary that would make my life easier?

N8G said:
the integrals for part 2 and 3 both turned out to be horrendous
They shouldn't. What do you get?

## What is the velocity of a football with and without drag?

The velocity of a football with and without drag depends on several factors such as the force applied, the mass of the football, and the air resistance. Without drag, the velocity of a football will continue to increase as long as a constant force is applied. However, with drag, the velocity will eventually reach a maximum due to the opposing force of air resistance.

## How does air resistance affect the velocity of a football?

Air resistance, also known as drag, is the force that opposes the motion of an object through air. As the football moves through the air, it creates a wake or disturbance in the air which results in drag. This drag force acts in the opposite direction of the motion of the football, therefore reducing its velocity.

## What factors affect the amount of drag on a football?

The amount of drag on a football is affected by its shape, size, and surface texture. A smooth, streamlined football will experience less drag compared to a rough, irregularly shaped football. Additionally, the density and viscosity of the air also play a role in determining the amount of drag on a football.

## How can the velocity of a football be calculated with and without drag?

The velocity of a football with and without drag can be calculated using the principles of Newton's Second Law of Motion. By considering the mass of the football, the force applied, and the amount of drag, the velocity can be determined using the equations of motion.

## How can the velocity of a football be measured in real-world scenarios?

In real-world scenarios, the velocity of a football can be measured using instruments such as radar guns or high-speed cameras. These devices can accurately measure the velocity of a football at different points in its flight and can also take into account the effects of drag. Additionally, the velocity of a football can also be estimated using mathematical models and simulations.

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