# Projectile Motion with Air Resistance

1. Mar 21, 2009

### glenn101

Hey all, I'm new to this forums.
My question is this, I've been learning about projectiles and I came across a thought regarding air resistance. For example if you have a more massive object (more kg) does it fall less distance(range) because of the addition of vectors of mg(weight force which would be bigger due to bigger m) and the air resistance force which will give a bigger net force? maybe I'm just confused.
Any help would be great

2. Mar 21, 2009

### arildno

Hi, glen101!

Let us assume that two projectiles, A and B are equal in shape, so that BOTH experience an air resistance force of type, say, $-k\vec{v}$, i.e, the constant k-value is equal for both projectiles.

Now, let us say that the mass of A is bigger than that of B.

Thus, Newton's 2.law for both objects can be written as:
$$\vec{g}-\frac{k}{m_{A}}\vec{v}=\vec{a}_{A}$$
$$\vec{g}-\frac{k}{m_{B}}\vec{v}=\vec{a}_{B}$$
$\vec{g}$ being the acceleration due to gravity.
Now, if A and B at some instant (say, initially), have the SAME velocity, it follows that the acceleration term induced by the air resistance is LESS on the more massive object A, because $\frac{k}{m_{A}}$ is less than $\frac{k}{m_{B}}$.

Thus, the heavier object should travel the furthest distance when beginning from equal conditions.

Last edited: Mar 21, 2009
3. Mar 21, 2009

### glenn101

I am intrigued that by what you are saying suggests that the more massive object should travel the furtherest distance as I have conducted an experiment launching a mass with a projectile launcher, and by my results it has shown that the more massive object travels less distance? would this be because the projectile launchers spring can only produce a certain amount of elastic potential energy to the mass thus the bigger mass consumes more energy during its flight making it move a shorter distance? oh and with the experiment I think the launch height and landing height are a bit different, not sure of that makes much of a difference however.

4. Mar 21, 2009

### Cantab Morgan

Putting it a bit whimsically, it's harder for air resistance to slow down the more massive object, because it has more momentum. (Assuming equal initial speeds)

Of course, it would take more energy to get the more massive object accelerated up to the same speed as the lighter one. It's interesting to muse about how air resistance will affect the range of the projectile if they are given identical initial energies instead of velocities.

5. Mar 21, 2009

### rcgldr

It's not a fair comparason, the spring won't launch the more massive object with the same speed as it would the less massive object.

6. Mar 21, 2009

### glenn101

Thank you all for the replies.
And you are very right Jeff Ried, I see the flaw in the experiment. And so if I had perfect conditions I would see that the more massive objects travels further distance due to the air resistance effect.

7. Mar 22, 2009

### csprof2000

Also, you must be careful. When you are doing projectile motion with air resistance, the optimal angle is no longer 45 degrees w.r.t. the horizontal.

8. Mar 22, 2009

### glenn101

@ csprof2000
yes that is true, also having the difference of launch height and landing height alters 45 degrees as the optimum angle. I have stated that in a report I am writing.

9. Mar 22, 2009

### csprof2000

Well, you should do more than state it... you should compare the two masses based on optimal trajectories, no? Each will have a different one.