Need help with Momentum Principle problem

[fillipeano]

Homework Statement

Suppose you are navigating a spacecraft far from other objects. The mass of the spacecraft is 3.0 x 10^4 kg (about 30 tons). The rocket engines are shut off, and you're coasting along with a constant velocity of ‹ 0, 22, 0 › km/s. As you pass the location ‹ 7, 7, 0 › km you briefly fire side thruster rockets, so that your spacecraft experiences a net force of ‹ 8.0 x 10^5, 0, 0 › N for 22.5 s. The ejected gases have a mass that is small compared to the mass of the spacecraft . You then continue coasting with the rocket engines turned off. Where are you an hour later? (Think about what approximations or simplifying assumptions you made in your analysis. Also think about the choice of system: what are the surroundings that exert external forces on your system?)

Homework Equations

Δp = ƩFΔt
rFinal = rInitial + (pFinal/mass)Δt

The answer is in meters.

The Attempt at a Solution

The hint I was given said this: Apply one step of the Momentum Principle, then one step of the position update equation using the new velocity just after the short burn of the thruster rockets. At this low speed the momentum is approximately m.

pFinal = <18.0 x 10^6, 6.6 x 10^5, 0>
so my final position equation looks like this:
<7,7,0> + <2160000,79200,0>

But my answer is wrong. Please help!

 

[gneill]

pFinal = <18.0 x 10^6, 6.6 x 10^5, 0>
so my final position equation looks like this:
<7,7,0> + <2160000,79200,0>

Recheck the calculation of your pFinal, paying particular attention to the powers of ten. Could be that you've been bitten by a unit conversion somewhere along the line.

 

[fillipeano]

Recheck the calculation of your pFinal, paying particular attention to the powers of ten. Could be that you've been bitten by a unit conversion somewhere along the line.

I checked, my answer is still wrong ):

 

[gneill]

I checked, my answer is still wrong ):

Well, let's go through your calculation step by step. Suppose you start by presenting your calculation for the change in momentum, Δp.

 

[fillipeano]

Okay, for delta p I did force x time, I get 18000000

 

[gneill]

Okay, for delta p I did force x time, I get 18000000

Okay. That's 1.8 x 10⁷ and it looks good. You should specify that it is a change in the x-component of the momentum (Momentum is a vector quantity). The units, of course, will be kg*m/s.

You can calculate the new velocity as v_1 = v_o + \frac{\Delta p}{M}
where v₁, v_o and Δp are all vectors. What do you get for the components of v₁?

 

[fillipeano]

Converting 22km/s into m/s I get 22000 m/s. Delta p/m = 600.

So, <0,22000,0> m/s + 600 kg m/s = <0,22600,0> kg m/s

 

[gneill]

Converting 22km/s into m/s I get 22000 m/s. Delta p/m = 600.

So, <0,22,0> m/s + 600 kg m/s = <0,22600,0> kg m/s

The force applied has only an x-component, so the Δp should have only an x-component, and the thus the velocity change should also be to the x-component. You've added the velocity change to the y-component.

 

[fillipeano]

Whoops, my bad.
<600,22000,0> for velocity.

 

[gneill]

Whoops, my bad.
<600,22000,0> for velocity.

That's better . Now, how about the position after 1 hour? Keep in mind that the initial position is specified in km (and likely the final answer is supposed to be in km units also).

 

[fillipeano]

Do I use rInitial + (pFinal/mass)*delta t?

 

[gneill]

r_f = r_i + v*t

Just like in 1D kinematics, only in this case the variables (except for time) are vectors.

You calculated v above. t and r_i are given in the problem statement.

 

[fillipeano]

I'll convert my initial position to meters since the answer requires it in meters
rfinal = <7000,7000,0> + <600,22000,0>*3600s = <2167000, 29000, 0>

 

[gneill]

I'll convert my initial position to meters since the answer requires it in meters
rfinal = <7000,7000,0> + <600,22000,0>*3600s = <2167000, 29000, 0>

Your y-component is not correct; you failed to multiply the y-velocity component by the time.

 

[fillipeano]

Again, my bad ):
rfinal = <2167000,79207000,0>

 

[gneill]

Again, my bad ):
rfinal = <2167000,79207000,0>

Okay, that looks good. Be sure to include the units on any results that you submit!

 

[fillipeano]

Okay, thank you so much for your help! My problem was that I was not using rFinal = rInitial + v*t correctly!