• Support PF! Buy your school textbooks, materials and every day products Here!

Variable acceleration and terminal velocity

  • Thread starter abdo799
  • Start date
  • #1
169
4

Homework Statement



A cube is falling from a height of 200 m (initial velocity =0) , it reaches it's terminal velocity . Given that :its mass is 100 kg , cross sectional area 40 cm2 and coefficient of drag 2. Putting only weight and drag in consideration , calculate its terminal velocity and time of the fall (g=10)

Homework Equations



W=mg , Fd= Cd*A*1/2*ρ*v2 ( ρ
=1)

The Attempt at a Solution


it tried to solve it using calculus , but all the formulas i had was integration or differentiation in respect to time , and i didnt have a function of time . i figured out a relation between velocity and acceleration , Fr=W-FD, after working it out , it was something like this , a=(100)/(1000-(0.08 v2) , i don't know what to do next , i dont know if i can substitute v with some something to get a function time
 
Last edited:

Answers and Replies

  • #2
199
15
it tried to solve it using calculus , but all the formulas i had was integration or differentiation in respect to time , and i didnt have a function of time . i figured out a relation between velocity and acceleration , Fr=W-FD, after working it out , it was something like this , a=(100)/(1000-(0.08 v2) , i don't know what to do next , i dont know if i can substitute v with some something to get a function time
$$a=\frac{dv}{dt}=\frac{dv}{dt}.\frac{dx}{dx}=\frac{dv}{dx}.\frac{dx}{dt}=\frac{dx}{dt}.\frac{dv}{dx}=v.\frac{dv}{dx}$$

Now, you have relation with which you don't need a function of time. And acceleration is a function of velocity.

Edit: check your acceleration again, it should be a little different.
 
  • #3
1,540
134

Homework Statement



A cube is falling from a height of 200 m (initial velocity =0) , it reaches it's terminal velocity . Given that :its mass is 100 kg , cross sectional area 40 cm2 and coefficient of drag 2. Putting only weight and drag in consideration , calculate its terminal velocity and time of the fall (g=10)

Homework Equations



W=mg , Fd= Cd*A*1/2*ρ*v2 ( ρ
=1)

The Attempt at a Solution


it tried to solve it using calculus , but all the formulas i had was integration or differentiation in respect to time , and i didnt have a function of time . i figured out a relation between velocity and acceleration , Fr=W-FD, after working it out , it was something like this , a=(100)/(1000-(0.08 v2) , i don't know what to do next , i dont know if i can substitute v with some something to get a function time
Recheck value of Fd.

Your expression of net acceleration also looks incorrect .

mdv/dt = Fd-mg .

Substituting value of Fd ,you will get mdv/(Fd-mg) =dt .

Integrate with proper limits and you will get velocity as a function of time.What do you get ?
 
Last edited:
  • #4
199
15
Integrate with proper limits and you will get velocity as a function of time.What do you get ?
He'll need to calculate velocity as a function of distance first, otherwise "velocity as a function of time" information would be useless.

But yes, he does need to calculate "velocity as a function of time".
 
  • #5
169
4
$$a=\frac{dv}{dt}=\frac{dv}{dt}.\frac{dx}{dx}=\frac{dv}{dx}.\frac{dx}{dt}=\frac{dx}{dt}.\frac{dv}{dx}=v.\frac{dv}{dx}$$

Now, you have relation with which you don't need a function of time. And acceleration is a function of velocity.

Edit: check your acceleration again, it should be a little different.
okay , this is how i got a
F= m/a
100/a = (100*10)-(1/2 * 1 * 2 * (0.42) v2)
100/a= 1000- (0.16 v2)
a=100/(1000-0.16v2)
if it's wrong please correct me
 
  • #6
1,540
134
He'll need to calculate velocity as a function of distance first, otherwise "velocity as a function of time" information would be useless.

But yes, he does need to calculate "velocity as a function of time".
I believe distance doesn't play a role in calculating terminal velocity ,unless you want to determine if the object achieves terminal velocity before reaching the ground.
 
  • Like
Likes 1 person
  • #7
199
15
okay , this is how i got a
F= m/a
100/a = (100*10)-(1/2 * 1 * 2 * (0.42) v2)
100/a= 1000- (0.16 v2)
a=100/(1000-0.16v2)
if it's wrong please correct me
Hey, ##F=m*a##.

Tanya already posted about this!!!.....understanding the situation is more important. But here is the result:

taking +ve direction to be downwards, we have:

$$mg-F_{d}=ma$$
$$a=g-\frac{F_{d}}{m}$$
Substitute the correct values, you'll get acceleration as a function of velocity.
 
  • Like
Likes 1 person
  • #8
199
15
I believe distance doesn't play a role in calculating terminal velocity ,unless you want to determine if the object achieves terminal velocity before reaching the ground.
terminal velocity is just the condition when the object is in equilibrium. ##F_{d}=mg## for the terminal velocity in this question. But the question asks the time of fall, not the time to reach terminal velocity. Time of fall is surely enough, dependent on the distance.

I hope you get the point!!!

Edit: Yes, distance doesn't play important role here to calculate terminal velocity. But I was talking about velocity in my previous posts. And its time of fall depends on velocity and not terminal velocity. But to calculate final velocity and hence calculate time, we need "velocity as a function of distance"...................to reach 100% of terminal velocity you need infinite/very large time, because velocity reaches terminal velocity asymptotically with time.
 
Last edited:
  • #9
169
4
Hey, ##F=m*a##.

Tanya already posted about this!!!.....understanding the situation is more important. But here is the result:

taking +ve direction to be downwards, we have:

$$mg-F_{d}=ma$$
$$a=g-\frac{F_{d}}{m}$$
Substitute the correct values, you'll get acceleration as a function of velocity.
worst mistake i've ever made , sorry , i drank like 5 cups of coffee , didnt sleep , got a test in 2 days
 
  • #10
199
15
worst mistake i've ever made , sorry , i drank like 5 cups of coffee , didnt sleep , got a test in 2 days
get some sleep, sleepiness and physics don't go hand in hand!!!!:tongue:
 
  • #11
169
4
get some sleep, sleepiness and physics don't go hand in hand!!!!:tongue:
Will do, thanks
 
  • #12
199
15

Related Threads on Variable acceleration and terminal velocity

Replies
9
Views
3K
Replies
12
Views
3K
Replies
4
Views
2K
  • Last Post
Replies
1
Views
10K
  • Last Post
Replies
1
Views
2K
  • Last Post
Replies
2
Views
2K
  • Last Post
Replies
1
Views
882
  • Last Post
Replies
10
Views
3K
  • Last Post
Replies
2
Views
4K
  • Last Post
Replies
3
Views
2K
Top