Question

Can someone show me the steps for this? (calc). The velocity of a particle moving on a line is v=ds/dt= 6 sin (2t) m/sec for all t. If s=0 when t=0, find s when t= pi/2 secs.

Answer 1

@johnny101
v = ds/dt
can we write
ds = vdt ?????

Answer 2

ds = 6 sin(2t) dt
is it possible for u do the integration of ds?

Answer 3

by using pi/2 as the upper and 0 as the lower?

Answer 4

no need to apply limits, first do the integration as an indefinite integral
we will use the boundary condition after determining s

Answer 5

can you help me with that im sorry im rusty on integrals

Answer 6

\[s =6* \frac{ \cos(2t) }{ 2 } + constant\]
first boundary condition is
s = 0 when t = 0 , hope u can determine the constant now

Answer 7

so for that setup you just did the deriv of sin? or did you use x^n+1/n +1? trying to follow the reasoning here..what you have is just 0

Answer 8

i did the integration of sinx
it is -cosx
i forgot to put the negative sign
s = 0 when t=0
\[0 = -6 * \frac{ \cos(0) }{ 2 } + constant\]
0 = -3 +constant as cos(0) = 1
therefore ,constant = 3
hence
\[s= -6*\frac{ \cos(2t) }{ 2 } + 3\]
now just put t = pi/2

Answer 9

did u follow the steps?

Answer 10

ok so sin -> -cos x i got that, where did the formation come from of -6* cos(0)/2 +C?

Answer 11

we get
\[s = -6*\frac{ \cos(2t) }{ 2 } + constant\]
when we solve an indefinite integral we always get some constant term
now the condition is given
s = 0 when t = 0
i just plug in the value of s and t which is 0

Answer 12

right, im saying, we have 6 sin 2t right? so you integrated which made sin -> -cos, which you then applied to the 6 right? so -6 * cos(2t), where is the / 2 coming from? from basic integration x^n+1/n+1? wouldn't that give 3? lost on the denominator i know its somethhing basic.

Answer 13

2 in the denominator has come from the chain rule of integration

Answer 14

can you show that please?

Answer 15

\[s = -6*\frac{ \cos(2t) }{ \frac{ d }{ dt }(2t) } + constant\]
\[s = -6*\frac{ \cos(2t) }{ 2 } + constant\]

Answer 16

i have the chain rule in my notes involving taking the exponent to the front, subtracting the exponent by one and multiplying by the deriv of what's inside...
so on this we have 6sin 2t. How did you convert that to 2t in fraction form? can you list it out(im sorry not grasping it from that directly)

Answer 17

this is the way of solving an integral using chain rule
we always take the derivative and it should be in the denominator
this is the rule , mate

Answer 18

so 2t is just the derivative of 6 sin 2t which is where that came from?

Answer 19

derivative in the denominator is a rule
u have to remember it
for example u need to determine integral of sin(5t+1)
we will get a solution like this
\[\frac{ -\cos(5t+1) }{ \frac{ d }{ dt }(5t+1)}\]
\[\frac{ -\cos(5t+1) }{ 5 }\]

Answer 20

ok so, 6 sin2t we integrate that which makes it -cos for the sin, with the negative transferring to the 6, so -6* cos 2t, whose deriv is 2t, so -6*cos2t/2t

Answer 21

\[-6*\frac{ \cos(2t) }{ \frac{ d }{ dt }(2t) }\]
u need to take the derivative of 2t

Answer 22

so we just take the 2t and put it in both the numerator and denominator, the deriv of 2t is just 2, which is why i wasnt following

Answer 23

\[ \text{ Let } a \text{ and } b \text{ be constants }\]
\[\int\limits_{}^{}b \sin(at ) dt \text{ where } a \neq 0\]
Let us do a substitution \[at=u => a dt =du \]
So we have in terms of u the integral is:
\[\int\limits_{}^{} b \sin(u) \frac{1}{a} du \text{ since } dt=\frac{1}{a} du\]
\[\text{ now } \frac{b}{a} \text{ is a constant } \]
so we have
\[\frac{b}{a} \int\limits_{}^{}\sin(u) du=\frac{b}{a} (-\cos(u))+C =\frac{-b}{a}\cos(u)+C\]
So the final answer in terms of t is
\[\frac{-b}{a} \cos(at)+C \text{ since } u=at \]
Now this is what I assumed you were pm-ing me about...Maybe.

Answer 24

well i was referring to the setup of it but yes. so based off what we were using before, that gives us with -6*cos 2t/2 + C,

Answer 25

after that i then..?

Answer 26

it says when t=0 s=0
so they are giving you info to find your C

Answer 27

once you find your C you have one more step

Answer 28

if i just input t=0 into that that gives me the 0 back, so i input pi/2 into that? so -6*cos(2pi/2)/2+C?

Answer 29

You said s(t)=-3 cos(2t)+C

Answer 30

Now it tells you when s=0 t=0
use that to find C

Answer 31

Then we can do the last step

Answer 32

Replace t with 0 and replace s with 0 like this
0=-3cos(2*0)+C
solve this equation for C

Answer 33

C=cos no since the 0 multiplies out to just 0 and 0/-3 = 0

Answer 34

so cos(0)=1 correct?

Answer 35

maybe you can solve this equation
0=-3+C

Answer 36

c =3, but how did you conclude cos(0)=1 if that solves to 0?

Answer 37

unit circle

Answer 38

ok roger

Answer 39

So we have s=-3 cos(2t)+3

Answer 40

now figure out what S is when t=pi/2

Answer 41

do you know what cos(pi) is?

Answer 42

-1