A projectile is fired vertically upward. Its distance s (in m) above the ground is given by s = 244t - 4.9t2, where t is the time (in s). Find the acceleration of the projectile.

Question

anonymous · Answer

Is everyone as lost as I was on this one?

amistre64 · Answer

im not

anonymous · Answer

could you explain please?

amistre64 · Answer

take 2 derivatives

amistre64 · Answer

the derivative of position is velocity, the derivative of velocity is acceleration

amistre64 · Answer

also, assuming this is shot on earth, the only force acting appreciably on a free "falling" object is gravity .... what is the acceleration of gravity?

anonymous · Answer

Not sure. Doesn't give it in the equation

amistre64 · Answer

oh, it gives it in the equation.

amistre64 · Answer

do you know how to take derivatives?

anonymous · Answer

Im no expert. We just started yesterday

amistre64 · Answer

well, polynomials are as simple as they get for derivatives; its just an application if what they call a power rule and a constant rule.

amistre64 · Answer

$$D_x[x^n]=x^{n-1}$$$$D_x[k]=0$$

amistre64 · Answer

ugh, forgot to type in the n ...
$$D_x[x^n]=n~x^{n-1}$$
thats better

anonymous · Answer

n being the?

amistre64 · Answer

the exponent of course, thats why its tucked up there in the corner of the x .... its the "power" of x .... hence the name, "power rule"

anonymous · Answer

Got it

amistre64 · Answer

so knowing these properties, how would you attempt a first derivative of the given setup for "s" ?

anonymous · Answer

f(x)  244t - 4.9t2

amistre64 · Answer

thats the original setup yes, but if you want to refer to it as a function of "x", you might wanna rewrite the ts as xs

anonymous · Answer

sorry thats not right

anonymous · Answer

s = 244x - 4.9x2?

amistre64 · Answer

if rewriting the ts as xs makes you more comfortable, thats fine; but we will still need to take a derivative at some point :)

244x  will derive to ______ ?

amistre64 · Answer

it might help to see this as:$$244~x^1$$

anonymous · Answer

I seem to be lost, but would it not derive to 244?

amistre64 · Answer

it would :)  and the derivative of a constant is 0, so to recap that term:

244x  derives once to 244, and again to 0

amistre64 · Answer

now lets try the second term:

-4.9t^2  whats the first derivative?

anonymous · Answer

Okay I figured I was wrong for getting the same answer :/

anonymous · Answer

-9.8t?

amistre64 · Answer

correct, and a derivative on that would get us?

anonymous · Answer

Thats where Im still a little confused. Like prior how you went from 244 to 0

amistre64 · Answer

all we did was take the derivative of a constant:

or to put it in power term:

244 x^0  derives to 244(0)x^(-1) = 0

anonymous · Answer

I got 0 again

anonymous · Answer

sorry computer froze for a minute

amistre64 · Answer

yeah, the site gets loopy ....

amistre64 · Answer

$$s=244~t^1-4.9~t^2$$
$$s'=244~(1)t^0-4.9~(2)t^1$$
$$s''=244~(1)(0)t^{-1}-4.9~(2)(1)t^0$$$$s'' = 0-9.8=-9.8$$

amistre64 · Answer

the acceleration on a free falling object is caused by gravity; the acceleration of gravity on the earth is -9.8 $\frac{meters}{sec^2}$

which means that the force is downward at a magnitude of 9.8

anonymous · Answer

I see. That helped me see the bigger picture.

amistre64 · Answer

good luck ;)

anonymous · Answer

did it not ask the acceleration upward though?

anonymous · Answer

thank you for your help!

amistre64 · Answer

no, it asked for the acceleration of the projectile, once the bullet leaves the gun, the only force accelerating it is gravity

anonymous · Answer

I understand. Thanks again I know it took longer than expected :)

amistre64 · Answer

youre welcome :)