Projectile with Air resistance problem

[ZenchiT]

Homework Statement

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.

Homework Equations

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!

 

[Qwertywerty]

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

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.

 

[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!

 

[Qwertywerty]

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!

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 ?

 

[rude man]

a) I think I've got part (a) correct. I did g/19.6 = k, therefore constant k = 1/2

Afraid not. What are the units of kv? of g? Can you really equate g = kv?

b)
My attempt would be
mkv-mg=ma
kv-g=dv/dt

Again, your equation contains terms of unequal dimensions (units). Fix this first.
Then, reconsider the signs in your equation.

 

[ZenchiT]

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.

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

 

[ZenchiT]

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 ?

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?

 

[ZenchiT]

Bump

 

[Qwertywerty]

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

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

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?

Yes, that's correct !

 

[rude man]

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

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?

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

 

[rude man]

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

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

 

[ZenchiT]

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

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

 

[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?

 

[rude man]

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

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

 

[rude man]

I have had another attempt at question a). mg - mkv = ma. Is that correct?

No, you are still mixing units.

, therefore when terminal velocity =19.6, k in terms of m is equal to, k = (m(g-a))/19.6

what is a ?

 

[ZenchiT]

No, you are still mixing units.

what is a ?

a is acceleration

 

[rude man]

a is acceleration

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?

 

[ZenchiT]

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

 

[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

 

[ZenchiT]

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.

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?

 

[haruspex]

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

With regard to signs...
The question as set has the initial velocity as upwards. It does not specify whether the equation you are to provide is to cover only that phase or whether it should be valid for all time. Think about the directions of gravity and drag in the upward and downward cases. What is different? How can you deal with that?

With regard to dimensions...
You are given that air resistance is kv. Is resistance a force or an acceleration?

 

[rude man]

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?

What are the units of kv? That is your first task now.
v = g/k is not a dimensionally consistent equation. And the exponent has to be dimensionless. kt is not dimensionless.

 

[ZenchiT]

I believe I've finally got the answer to part a)!
F=ma and F=mg (gravity acting downwards) - kv (resistive force)
Therefore...
mg-kv=ma
Rearranging, where v=19.6
k= m(g-a)/19.6

Is that correct?!

 

[ZenchiT]

Please tell me that's correct!

 

[haruspex]

I believe I've finally got the answer to part a)!
F=ma and F=mg (gravity acting downwards) - kv (resistive force)
Therefore...
mg-kv=ma
Rearranging, where v=19.6
k= m(g-a)/19.6

Is that correct?!

Almost right. You are told the teminal velocity. What value does that imply for a?

 

[ZenchiT]

Almost right. You are told the terminal velocity. What value does that imply for a?

That it is equal to 0?! So we have 9.8m/19.6=k, therefore k=m/2?!

Thank you very much for your help? Any chance I can get the solution for part b), need it really urgently :/

 

[haruspex]

That it is equal to 0?! So we have 9.8m/19.6=k, therefore k=m/2?!

Thank you very much for your help? Any chance I can get the solution for part b), need it really urgently :/

You cannot write k = m/2 because that is only true in a specific system of units. When you plug in the values for g and v, you should plug in the units too. There will be some cancellattion, yielding a set of units to apply to m/2. (Just treat the units, like kg, m, s as though they are algebraic variables.)

For part b, no we do not simply give solutions, no matter how urgent. You should now be able to write the equation relating forces and acceleration.

 

[Qwertywerty]

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

We are only discussing the case in which the ball goes up.

 

[haruspex]

We are only discussing the case in which the ball goes up.

Not exactly. The terminal velocity is given, and a deduction is to be made from it. That implies considering the downward stage.

 

[Qwertywerty]

Not exactly. The terminal velocity is given, and a deduction is to be made from it. That implies considering the downward stage.

I was discussing part b.

 

[ZenchiT]

Guys! I think I've got the answer, I'm just going to upload it now. PLEASE BE RIGHT!

 

[ZenchiT]

Part a) F=mg and F=kv

Therefore mg=kv and v=19.6 (terminal velocity)
k = mg/v
k = mg/19.6

Part b)

Equation of motion -> -mg-kv = ma
(Sub in k =mg/v)
-2mg = ma
a = -2g
dv/dt = -2g
v = -2gt

Or maybe instead of subbing in k=mg/v It should have been mg/19.6?

Voila! Surely that's correct right?!

 

[haruspex]

Part a) F=mg and F=kv

Therefore mg=kv and v=19.6 (terminal velocity)
k = mg/v
k = mg/19.6

Part b)

Equation of motion -> -mg-kv = ma
(Sub in k =mg/v)
-2mg = ma
a = -2g
dv/dt = -2g
v = -2gt

Or maybe instead of subbing in k=mg/v It should have been mg/19.6?

Voila! Surely that's correct right?!

For the value of k, as I mentioned, if you are going to put numbers representing dimensioned quantities in the answer then you also need units. You don't need units for m and g since symbolic variables encapsulate the physical quantity independently of units. So you just need the right units for the 1/19.6 factor. What would those be?
For part b), you have used the same symbol v for two different velocities, the terminal velocity and the initial velocity. So, yes, you should have substituted mg/19.6 (with units) for k.

 

[ZenchiT]

For the value of k, as I mentioned, if you are going to put numbers representing dimensioned quantities in the answer then you also need units. You don't need units for m and g since symbolic variables encapsulate the physical quantity independently of units. So you just need the right units for the 1/19.6 factor. What would those be?
For part b), you have used the same symbol v for two different velocities, the terminal velocity and the initial velocity. So, yes, you should have substituted mg/19.6 (with units) for k.

Would it have N as the units because it has the same dimensions as force?

 

[haruspex]

Would it have N as the units because it has the same dimensions as force?

The problem is the lack of units for the 19.6. What units should that have? So what units will the 1/19.6 have?