Arithmetic sequence: We throw an object from the hight of 12,261 meters.

[Stripey]

Hi,
I need help with solving this problem:
We throw an object from the hight of 12,261 meters. In the first second it is travelling with the speed of 4.904 m/s and accelerating by 9.808 m/s every second. How long will it take for the object to hit the ground?

I figured out we can use the basic formula for arithmetic sequence where a₁=4.904 and d=9.808. This way we get a_n=9.808*n - 4.904 but I wasn't 100 % sure how to continue. I thought that Sn might be 12,261 and then used the sum formula to calculate that the time (represented by n) would be 50.002039 seconds.
Is this approach and result correct?

Thanks in advance for your help.

 

[lev888]

This is not an arithmetic problem. Look up motion formulas - i.e. distance traveled as a function of time, initial speed and acceleration.

 

[Dr.Peterson]

Hi,
I need help with solving this problem:
We throw an object from the hight of 12,261 meters. In the first second it is travelling with the speed of 4.904 m/s and accelerating by 9.808 m/s every second. How long will it take for the object to hit the ground?

I figured out we can use the basic formula for arithmetic sequence where a₁=4.904 and d=9.808. This way we get a_n=9.808*n - 4.904 but I wasn't 100 % sure how to continue. I thought that Sn might be 12,261 and then used the sum formula to calculate that the time (represented by n) would be 50.002039 seconds.
Is this approach and result correct?

Thanks in advance for your help.

Properly speaking, this requires (basic) calculus. If you are right about the way you are intended to solve it, then you are modeling the motion by imagining that it goes at a constant speed each second, and then instantaneously speeds up by a fixed amount. That may be what they are saying, or it may not. It is definitely not the way physics really works, but it can be a way to begin moving toward calculus.

Can you tell us the context of the question? What class are you taking, and what have you recently learned that this would be testing? Have you been given any examples that would suggest using your method (arithmetic series), or did you get that idea on your own?

 

[HallsofIvy]

Hi,
I need help with solving this problem:
We throw an object from the hight of 12,261 meters. In the first second it is travelling with the speed of 4.904 m/s and accelerating by 9.808 m/s every second. How long will it take for the object to hit the ground?

I figured out we can use the basic formula for arithmetic sequence where a₁=4.904 and d=9.808.

Unfortunately that is wrong. With acceleration -9.908 m/s^2, the velocity after t seconds is 4.904- 9.908t . The velocity, after t seconds, is an arithmetic sequence but the height is 12261+ 4.904t- 4.954t^2 (12261 meters? Were you in a helicopter or on the edge of a very highcliff?). Set that equal to 0 and and solve for t.

This way we get a_n=9.808*n - 4.904 but I wasn't 100 % sure how to continue. I thought that Sn might be 12,261 and then used the sum formula to calculate that the time (represented by n) would be 50.002039 seconds.
Is this approach and result correct?

Thanks in advance for your help.

 

[Dr.Peterson]

You called it an arithmetic sequence, but mentioned using the sum formula -- that is, the sum of the series. That is correct; but the sum (of the sequence you described, with a small fix) will give the total distance traveled, not the final height. If you do the work right, you will get the formula that was mentioned.

I notice that the sequence that was set up for you corrects for the discrete model you are using; the distance traveled in the first second is given as half the actual speed that would be attained at the end of that second, that is, the average speed over that interval. So it turns out that this model will give accurate results.

I would still like to hear more about the context, but I think you are doing the right things, apart from the details I mentioned, and possibly some issues with how you are indexing the series. To make sure, I'd like to see the details of your work.

 

[Stripey]

You called it an arithmetic sequence, but mentioned using the sum formula -- that is, the sum of the series. That is correct; but the sum (of the sequence you described, with a small fix) will give the total distance traveled, not the final height. If you do the work right, you will get the formula that was mentioned.

I notice that the sequence that was set up for you corrects for the discrete model you are using; the distance traveled in the first second is given as half the actual speed that would be attained at the end of that second, that is, the average speed over that interval. So it turns out that this model will give accurate results.

I would still like to hear more about the context, but I think you are doing the right things, apart from the details I mentioned, and possibly some issues with how you are indexing the series. To make sure, I'd like to see the details of your work.

It was a "bonus" question in a financial maths class so I don't think the physics of it really matters. That's why I used an arithmetic sequence which is what a lot of the formulas in that field are based on.
Sorry if this is a stupid question, but is there a difference between the total distance traveled and the final height in this case? I mean if we keep the problem simple and don't take into account anything like crosswinds.

 

[Stripey]

Unfortunately that is wrong. With acceleration -9.908 m/s^2, the velocity after t seconds is 4.904- 9.908t . The velocity, after t seconds, is an arithmetic sequence but the height is 12261+ 4.904t- 4.954t^2 (12261 meters? Were you in a helicopter or on the edge of a very highcliff?). Set that equal to 0 and and solve for t.

Could you please elaborate on how you came up with the velocity after t seconds and the other equation?

 

[Dr.Peterson]

It was a "bonus" question in a financial maths class so I don't think the physics of it really matters. That's why I used an arithmetic sequence which is what a lot of the formulas in that field are based on.
Sorry if this is a stupid question, but is there a difference between the total distance traveled and the final height in this case? I mean if we keep the problem simple and don't take into account anything like crosswinds.

When you drop something, the final height is the initial height minus the distance traveled. Going so far down reduces your altitude by so much.

As I mentioned in my later comment, the problem is set up to do all the physics for you, so that the arithmetic series would do the job. It probably would have helped if you had told us the context from the start, so we'd know what to focus on, e.g. that you don't have any calculus (which is what HallsofIvy used to get that formula, though I think you'll end up with the same formula using the series).

 

[Stripey]

When you drop something, the final height is the initial height minus the distance traveled. Going so far down reduces your altitude by so much.

As I mentioned in my later comment, the problem is set up to do all the physics for you, so that the arithmetic series would do the job. It probably would have helped if you had told us the context from the start, so we'd know what to focus on, e.g. that you don't have any calculus (which is what HallsofIvy used to get that formula, though I think you'll end up with the same formula using the series).

HallsofIvy's result is only 0.2 from mine so as you say, both should be (at least roughly) correct. Thank you