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?423 2 tvst
by given isvelocity its if at object an of position the is What
1 Example
Remember that given a position function we can find the velocity by evaluating the derivative of the position function with respect to time at the
particular time.
dt
xdtvtx
then , Given
Hence, what we need is an antiderivative of the velocity function.
dtvx
dttx 42 2
ttx 43
2 3
?423 2 tvst
by given isvelocity its if at object an of position the is What
1 Example
ttx 43
2 3
To check our answer we need only take the derivative of the result.
dt
xdv
tt
dt
dv 4
3
2 3
42 2 tv
It’s appears we have found the right function …
or have we????
?423 2 tvst
by given isvelocity its if at object an of position the is What
1 Example
ttx 43
2 3
What if the position function is
543
2 3 ttdt
dv
42 2 tv
543
2 3 ttx
So this function also works!
?423 2 tvst
by given isvelocity its if at object an of position the is What
1 Example
ttx 43
2 3
What if the position function is
1043
2 3 ttdt
dv
42 2 tv
1043
2 3 ttx
So this function works as well!
?423 2 tvst
by given isvelocity its if at object an of position the is What
1 Example
constant re whe CCttx 43
2 3Since the derivative of a constant is 0, it seems that any function of the form
works just as well!
constantany is where
Integral Indefinite
CCxFdxxf
?423 2 tvst
by given isvelocity its if at object an of position the is What
1 Example
msx Suppose 50 In order to find a particular solution you must be given an initial condition.
Cmx 0403
25)0( 3
mC 5
543
2 3 ttx
Particular Solution
53433
23 3 sx
msx 353
Cttx 43
2 3
? if
by given is onaccelerati whose at object an ofvelocity the is What
2 Example
smsvt
smt
sma
st
.90466
2
243
dtav
dtttv 266
Cttv 32 23
Csmsv 32 4243.904
But
smC .10
.1023 32 ttv
? if
by given is onaccelerati whose at object an ofvelocity the is What
2 Example
smsvt
smt
sma
st
.90466
2
243
.1023 32 ttv
.1022232 32 v
smv 142
.000.20.30.2 2 mxtttv and by given is particle a ofvelocity The
3 Example
? at position sparticle' the is Whata. st 0.3
dtvx
dtttx 232 2
Ctttx 22
3
3
2 23
Cx 0202
30
3
200 23
But
0C
tttx 22
3
3
2 23
.000.20.30.2 2 mxtttv and by given is particle a ofvelocity The
3 Example
? at position sparticle' the is Whata. st 0.3
tttx 22
3
3
2 23
0.320.32
30.3
3
20.3 23 x
mx 5.10.3
.000.20.30.2 2 mxtttv and by given is particle a ofvelocity The
3 Example
? at onaccelerati sparticle' the is Whatb. st 0.3
dt
vda
232 2 ttdt
da
34 ta
30.340.3 a
20.90.3s
ma
.000.20.30.2 2 mxtttv and by given is particle a ofvelocity The
3 Example
? to from ntdisplaceme sparticle' the is Whatc. stst 0.40.2
42 dtvx
42
2 232 dtttx
4
2
23 22
3
3
2
tttx
222
2
32
3
2424
2
34
3
2 2323x
3
24
3
210x
mx 15
? at position its is What. position at speed withupward thrown is ballA
4 Example
1ttxv oo
Earth the of surface the near constant ga
dtgdta
Ctgv
Cgvv o 00
But
ovC
ovtgv 1st Kinematic Equation
? at position its is What. position at speed withupward thrown is ballA
4 Example
1ttxv oo
tgvv o 1st Kinematic Equation
dttgvdtv o
Ctgtvx o 2
2
1
Cgvxx oo 202
100
But
oxC
oo xtgtvx 2
2
1
? at position its is What. position at speed withupward thrown is ballA
4 Example
1ttxv oo
2
2
1tgtvxx oo
3rd Kinematic Equation
oo xtgtvx 2
2
1
2111 2
1tgtvxtx oo