# Homework Help: Projectile with Air resistance problem

1. Aug 25, 2015

### ZenchiT

1. The problem statement, all variables and given/known data
The resistance to motion of a projectile, of mass m, is proportional to its speed. When falling vertically its terminal velocity is 19.6 m/s. The projectile is thrown vertically upwards with gravity acting downwards.
(a) Assume that the resistance to motion can be expressed as kv where v is the velocity andk is a constant. Determine k in terms of m using the terminal velocity value 19.6 m/s.
(b) Assume that the projectile is launched vertically upwards with a speed of 6 m/s. Write down the equation of motion which describes the resulting motion and find the expression
for the velocity, v, at time t.

2. Relevant equations

3. The attempt at a solution

a) I think I've got part (a) correct. I did g/19.6 = k, therefore constant k = 1/2
b) My biggest problem with this question is working out the resultant force. The way I would attempt this question is work out the resultant force and then use the equation F=ma. I would substitute a=dv/dt and then simply solve the differential equation by making v the subject. However I can't seem to work out what F would be.
I know that you would have to subtract mg (weight) from the projectile force.
My attempt would be
mkv-mg=ma
kv-g=dv/dt

Thank you so much for your help!

2. Aug 25, 2015

### Qwertywerty

Is force of gravity upwards or downwards ? Does air resistance act upwards or downwards ?

You'll need to use a bit of calculus here.

Hope this helps.

3. Aug 25, 2015

### ZenchiT

The force of gravity would be downwards, so would mg be positive? And the air resistance would act upwards, so on the left handside you would have mkv+mg? And then you proceed to integrate it?

Thank you!

4. Aug 25, 2015

### Qwertywerty

No, your final step is correct, but your reasoning is not. If velocity is upwards, and air resistance opposes it, will it act upwards or downwards ?

5. Aug 26, 2015

### rude man

Afraid not. What are the units of kv? of g? Can you really equate g = kv?
Again, your equation contains terms of unequal dimensions (units). Fix this first.
Then, reconsider the signs in your equation.

6. Aug 26, 2015

### ZenchiT

I'm sorry I don't quite understand :/, if the dimensions are incorrect, what would I have to do?

7. Aug 26, 2015

### ZenchiT

Oh okay I think I get it! So if velocity is upwards then air resistance will act downwards, does that mean you have air resistance AND gravity opposing the velocity?

8. Aug 26, 2015

### ZenchiT

Bump

9. Aug 26, 2015

### Qwertywerty

In your original question, you have said that kv is a force. Does g represent force or acceleration ?
Yes, that's correct !

10. Aug 26, 2015

### rude man

No, that's incorrect.
If I drop something does gravity act to oppose velocity? Try jumping off a roof!

11. Aug 26, 2015

### rude man

Why not try Newton: sum of forces = mass times acceleration? And you need to decide what the signs of kv and mg are.

12. Aug 26, 2015

### ZenchiT

Haha! Now I'm super confused! I'm getting two different answers :/

13. Aug 26, 2015

### ZenchiT

I have had another attempt at question a). mg - mkv = ma, therefore when terminal velocity =19.6, k in terms of m is equal to, k = (m(g-a))/19.6

Is that correct?

14. Aug 26, 2015

### rude man

On PF, one of the things you have to decide is to whom to listen!

15. Aug 26, 2015

### rude man

No, you are still mixing units.
what is a ???

16. Aug 26, 2015

### ZenchiT

a is acceleration

17. Aug 26, 2015

### rude man

What acceleration? Accel varies from the moment the projectile is fired in the UP direction until terminal velocity is reached in the DOWN direction .. so what number do you plan to ascribe to a?

18. Aug 26, 2015

### ZenchiT

Ahh okay, would I be able to write it as dv/dt, and then solve to get velocity?

19. Aug 26, 2015

### ZenchiT

Can someone please run me through this, I'm more confused than ever! Sorry to be a pain :/

Thank you for all the support so far

20. Aug 26, 2015

### ZenchiT

In terms of unequal dimensions, I believe the dimensions are correct? Because the equation I have is v= (g/k)(1- e^-kt). where g is gravitational acceleration and k is a constant. Terminal velocity is when t tends to infinity. If t tends to infinity then e^-kt tends to 0. Therefore v = (g/k). Why isn't that correct?