Question

1) if an object of mass m has velocity v, then its kinetic energy K is given by k=1/2mv^2. if v is a function of time t, use the chain rule to find a formula for dK/dt.

2)When a space shuttle is launched into space, an astronaut’s body weight decreases until a state of weightlessness is achieved. The weight, W, of a 150-lb astronaut at an altitude of x kilometers above sea level is given by
W=150 (6400/ 6400+x)^2
If the space shuttle is moving away from the earth’s surface at a rate of 6 km/sec, at what rate is W decreasing when x = 1000 km? (Hint: Keep in mind that x is changing over time. You can think of x as x(t), a function of time.)

Answer 1

\[
{d K\over d t}=\left({1\over 2}m\right)(2v){d v\over d t}
\]

Answer 2

for 2, it is given that \[{d x\over d t}=-6{\rm km/s}\]the negative sign because it is moving away.
apply chain rule to find the derivative of "W"

Answer 3

that's what giving me problems. i have trouble applying the chain rule given that i don't quite understand it that well :/

Answer 4

try it. you take the derivative of functions till you cannot
\[W=150\left(6400\over6400+x\right)^2
\]
here first let us make \[X={6400\over6400+x}\\
W=150X^2
\]
what is the derivative of W now?

Answer 5

300x??

Answer 6

good.
\[W=300X\]
replace X and multiply with its derivative
\[
W=300\left(6400\over6400+x\right)\times{d\over dx}\left(6400\over6400+x\right)
\]

Answer 7

x would be 1000 km , right?

Answer 8

not yet. first evaluate the derivative

Answer 9

would the result be 300 (x+1)?

Answer 10

how?

Answer 11

math cannot be solved by guess work. it is very mechanical. just follow the steps. do not think.