rate of change question help plssss

[aadharmg]

A raindrop falls so that its velocity v m/s at time t seconds is given by dv/dt = [g - (v/10)].
(a) Find the terminal velocity of the raindrop.
(b) Find the velocity after 12 seconds.
(c) Find the height of the cloud where the raindrops were created if each raindrop takes 2 minutes to reach the ground.

[RuiAce]

I feel as though this is an MX2 question. Please provide the source if you need further assistance.

Some guidance:
a) t->inf or a->0
b) Find t as a function of v
c) Progress further and find x as a function of t

[aadharmg]

I feel as though this is an MX2 question. Please provide the source if you need further assistance.

Some guidance:
a) t->inf or a->0
b) Find t as a function of v
c) Progress further and find x as a function of t

This is actually a question from Terry Lee. All the questions in this exercise (Challenge Problems 9 in the MX1 book) are mind boggling and I can't figure them out after wasting a lot of time on them. This specific question is Question 1 from Challenge Problems 9.

[RuiAce]

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The rest is left as your exercise. For b), it's a simple matter of substituting \(t=10\) now.

For c), integrate to get x as a function of t. When t=0, x=H where H is the height, and when t=2, x=0.

If you're stuck, feel free to post further progress.

[aadharmg]

____________________________________




The rest is left as your exercise. For b), it's a simple matter of substituting \(t=10\) now.

For c), integrate to get x as a function of t. When t=0, x=H where H is the height, and when t=2, x=0.

If you're stuck, feel free to post further progress.

Thanks for the guidance, I was able to make an equation for x in terms of t. The problem now is that because t is in seconds, when x = 0, t must be = 120 because after 2 minutes it reaches the ground. To find the constant after integrating the velocity equation, when I substitute 120 for t and 0 for x, I get a really weird number. If I carry that number, when I substitute t = 0 to find H, I end up with a negative and bizarre answer.

[RuiAce]

Thanks for the guidance, I was able to make an equation for x in terms of t. The problem now is that because t is in seconds, when x = 0, t must be = 120 because after 2 minutes it reaches the ground. To find the constant after integrating the velocity equation, when I substitute 120 for t and 0 for x, I get a really weird number. If I carry that number, when I substitute t = 0 to find H, I end up with a negative and bizarre answer.

I think I realised the problem.

What usually happens in mechanics is that the orientation flips when you're considering upwards or downwards motion. I think in this scenario, you consider downwards as the positive motion.

So instead what you should be doing is treating the highest you can be (the cloud) the origin, i.e. x=0. So when t=0, x=0. Then when t=120, x=H, i.e. H is how far down you are from the cloud.

(Pardon the late reply - still doing stuff for lectures.)

[aadharmg]

I think realised the problem.

What usually happens in mechanics is that the orientation flips when you're considering upwards or downwards motion. I think in this scenario, you consider downwards as the positive motion.

So instead what you should be doing is treating the highest you can be (the cloud) the origin, i.e. x=0. So when t=0, x=0. Then when t=120, x=H, i.e. H is how far down you are from the cloud.

(Pardon the late reply - still doing stuff for lectures.)

Yessssss, I also finally understand. Thank you so much for the help, considering I have trials coming up, it's great that I can now tackle such a problem, or at least understand it. Thankkkkssssss! (PS the late replies weren't noticed at all, so it's all good)