Kirk's Mission: Solving for Acceleration Time

[fogvajarash]

Homework Statement

Captain Kirk is on an important space mission to bring a Klingon ambassador to space outpost 13. He must travel 6000. km in less than 30.0 min, or else war will be declared between the Klingons and humans. After traveling exactly 27.0 min at a steady pace, he sees that there are still 870.0 km left to go.
His ship must then accelerate at 20.000 m/s^-2 for how many seconds in order to achieve the desired time?

Homework Equations

s = u + at
s = v_ot + 1/2at²
v² = u² + 2as

The Attempt at a Solution

I've tried separating the motion of the spacecraft in 3 different times. Time 1 is when the spacecraft is at constant speed until 27.0min, time 2 is when the spacecraft starts to accelerate at 20.000m/s^-2 and time 3 is when the spacecraft travels at the new constant speed increased by the acceleration. So:

For time 1:

s = v_ot

Here i found the speed just by plugging in s = 5130000m and t = 1620s.

For time 2 and 3:

s = v_ot + 1/2at² + v_f(180-t)

180-t as the spacecraft has 3 minutes remaining (180s). I found the v_f with the following equations:

v_f = v₀ + at

I've tried this problem numerous times and I keep getting that the result is 0.044s, but it is wrong.

 

[HallsofIvy]

You are not taking into account the distance traveled while accelerating.
Yes, he must go 870 km in 3 m (870000 m in 180 s). If he accelerates at 20 m/s^2 for T seconds. he will have gone a distance $10T^2+ 3166.67 T$ and will have achieved a speed of 20T m/s. He must have $$10T^2+ 3166.67T+ (20T)(180- T)= 870$$.

(Knowing Kirk, he will probably dawdle in order to prolong the war!)

 

[mfb]

@HallsofIvy: He did that into account (at least in the equation s=...). I think you are mixing meters and kilometers here.

@fogvajarash: Here is a quick way to see that 0.044s cannot be right:
With an acceleration of 20m/s^2, accelerating for 0.044s changes the speed just by roughly 1m/s, from 3167 to 3168m/s. To travel the remaining 870km in 3 minutes, the average speed (which is lower than the maximal speed) has to be of the order of 4800m/s. The acceleration time has to be much more than 0.044s.

Please show your work. Your approach looks right, so we have no way to tell what you did wrong if you don't show it.

By the way: m/s^-2 does not make sense. ms^-2 = m/s² are correct units for an acceleration.

 

[fogvajarash]

@HallsofIvy: He did that into account (at least in the equation s=...). I think you are mixing meters and kilometers here.

@fogvajarash: Here is a quick way to see that 0.044s cannot be right:
With an acceleration of 20m/s^2, accelerating for 0.044s changes the speed just by roughly 1m/s, from 3167 to 3168m/s. To travel the remaining 870km in 3 minutes, the average speed (which is lower than the maximal speed) has to be of the order of 4800m/s. The acceleration time has to be much more than 0.044s.

Please show your work. Your approach looks right, so we have no way to tell what you did wrong if you don't show it.

By the way: m/s^-2 does not make sense. ms^-2 = m/s² are correct units for an acceleration.

Crap I meant 20000m/s2 instead of 20.0. Sorry for the mistake

 

[mfb]

That's still not enough, based on the same estimate.
Again, please show your work.

And that acceleration will crush every living object on the ship.

 

[fogvajarash]

That's still not enough, based on the same estimate.
Again, please show your work.

And that acceleration will crush every living object on the ship.

I suppose the problem isn't that realistic haha.

Here's my procedure:

 

[mfb]

There are 870km left, not 730km.

 

[fogvajarash]

There are 870km left, not 730km.

I get as a final result t = 0.0834s. Is this result accurate? (I've changed the 730000 by a 870000).

 

[mfb]

You can use this value and plug it in your initial equation for s= to check it. It looks reasonable, but I did not check it in detail (WolframAlpha can do that better anyway).

 

[fogvajarash]

You can use this value and plug it in your initial equation for s= to check it. It looks reasonable, but I did not check it in detail (WolframAlpha can do that better anyway).

I keep getting an incorrect result for the exercise. Have i made a mistake with setting up the equations? It looks alright to me.

 

[mfb]

The equations look good.
Are you sure the acceleration is 20000m/s^2? That number looks really odd compared to the other numbers, 20m/s^2 would give more reasonable results.