Question

Falling Ball:
When an object is allowed to fall freely near the surface of the earth, the gravitational pull is such that the object falls 16 ft in the first second, 48 ft in the next second, 80ft in the next second, and so on.
a) Find the total distance a ball falls in 6seconds
b) Find a formula for the total distance the ball falls in n seconds

Answer 1

@jtvatsim Whenever you're free! I'll try to work it out on my own until then :)

Answer 2

Sounds good, be back. :)

Answer 3

a_6=496feet

Answer 4

^ (part a)

Answer 5

b) the formula would be
a_n=16+(n-1)(32)

Answer 6

Testing it with a_2 the formula seems to be correct :)

Answer 7

I got the same thing for b, but 576 for a

Answer 8

hmm let's see where I might've messed up
a_6=16 +(6-1)(32) correct yeah?

Answer 9

to find the 6th term, yes. But I think it's asking for the sum of the first 6 terms

Answer 10

oooh, wow. I completely missed that.

Answer 11

so i must find a_4 and a_5

Answer 12

a_4=112, a_5 = 144

Answer 13

for this one it's easy enough to do that, but there's a formula to find the sum of the first n terms.\[S _{n}=\frac{ n }{ 2 }(a _{1}+a _{n})\]

Answer 14

oh o.0

Answer 15

Yes those are right

Answer 16

hm but I should probably use that formula you just used because the prof introduced it to us.

Answer 17

a_6=176 correct?

Answer 18

yes

Answer 19

so we have
s_6=(6/3)(16+176)
=3(192)
=576

Answer 20

yep

Answer 21

Fantastic! thank you so much :)

Answer 22

you're welcome!

Answer 23

oh wait! @peachpi the b) asks "TOTAL distance the ball falls at n seconds"

Answer 24

my formula is not for the sum

Answer 25

Right?
So the formula would actually be
s_n=(n/2)(16+a_n)

Answer 26

right, and you'd so substitute the formula for the nth term for a_n

Answer 27

If you need an explicit formula

Answer 28

ok, it just asks for a formula that could generate any nth term.

Answer 29

Er, should I put the formula to find
a_n
and also the formula for finding the sums?

Answer 30

so I think what they're asking is for a formula where you would just stick in the n value and get the sum.

Sn = 3(16 + a_n)
but since we know a_n = 16+(n-1)(32)

Sn = 3(16 + 16+(n-1)(32))

Answer 31

does that make sense? can't really think of a right way to say it

Answer 32

you mean
sn=n/6(16+(n-1)(32))
(not ^3)

Answer 33

ahh yes I see what you did.

Answer 34

oh, yes. n/2

Answer 35

ok :)

Answer 36

thanks, phew. That was close. I almost put the wrong formula xP

Answer 37

Sn = (n/2)(16 + 16+(n-1)(32))
is the final formula?

Answer 38

yes. I mean you can do some algebra to make it prettier, but it's correct

Answer 39

oh o.0 we should probably make it prettier.. hah

Answer 40

sn=(n/2)(16+512(n-1))
is that still correct? and simplified enough?

Answer 41

keep going until you get a quadratic, so distribute 512 and combine like terms

Answer 42

wait no!