Question

What is the final velocity of a rocket propelled into space with a net acceleration of 200m/s^2 as it reaches an altitude of 10.0km? (Assuming an initial velocity of zero)

***thank you!!

Answer 1

Is this kinematics, or energy?

Answer 2

more kinematics:)

Answer 3

First look at velocity: \[
v(t)-v(0) = \int_0^ta(x)~dx = \int_0^t200(m/s^2)~dx = 200(m/s^2)\bigg|_{x=0}^{x=t}
\]Initial velocity is zero, meaning \(v(0)=0\). We end up with:\[
v(t) = 200t(m/s^2)
\]

Answer 4

Now we need to find \(t_f\), which is the time it reaches that high altitude.

Answer 5

okay:)

Answer 6

\[
x(t_f)-x(0) = \int_0^{t_f}200t(m/s^2)~dt = 200\frac{t^2}{2}(m/s^2)\bigg|_{t=0}^{t=t_f}
\]We know that \(x(t_f)-x(0)=10.0(km)\) since that was its increase in altitude.
So we get: \[
10.0(km) =200\frac{(t_f)^2}2(m/s^2)
\]We can use this to solve for \(t_f\).

Answer 7

ohh okay, so from here, would I first multiply by 2 on both sides?

20.0 (km)=200(tf)^2 ?

Answer 8

Yeah

Answer 9

and then now, divide by 200?

0.1= tf^2 ?

Answer 10

and square and get tf=0.316 ?

Answer 11

Yes, but remember your units. \[
0.1(km) = t_f^2(m/s^2)
\]

Answer 12

ahh right!! okay, and then tf=0.316 km/s/s^2 ?

Answer 13

Hold on, what units should \(t_f\) be in?

Answer 14

m/s^2 ?

Answer 15

Nope.

Answer 16

ohh darn :( not sure!! km?

Answer 17

First divide both sides by the units. \[
\frac{0.10(km)}{(m/s^2)}=t_f^2
\]

Answer 18

okay, so 0.10 km = t^f(m/s^2) ?

Answer 19

Remember that \(1km = 1000m\)

Answer 20

so 100 m = tf^2(m/s^2) ?

Answer 21

Yeah \[
100m = t_f^2(m/s^2)
\]First divide by \(m\) to get: \[
100 = t_f^2(1/s)^2
\]Then multiply by \(s^2\) to get: \[
100(s^2)=(t_f)^2
\]Then you take the square root.

Answer 22

ahhh okie:)

and then we get tf=10 ?

Answer 23

\[
t_f=10(s)
\]

Answer 24

ohh okay:) and so that would be the final velocity? :O

Answer 25

Then we need to put that into our equation we got for velocity:\[
v(t) = 200t(m/s^2)
\]

Answer 26

\[
v_f=v(t_f) = 200(10(s))(m/s^2)
\]

Answer 27

oh! okay:)

and that's the final velocity? :O

plug in and done? :O

Answer 28

Simplify it.

Answer 29

so we get 2000 m/s ?

Answer 30

@wio? did i simplify that correctly? would 2000 m/s be the final velocity?

Answer 31

I think so.

Answer 32

okie!! yay! thank you very much!! :D