vector functions...help

[matpo39]

i need a little help with this problem, i can get most of it set up but there is one part where i get stuck heres the question.

You are the captin of a spaceship that has been given the mission of flying from a spacestation at position (1,2,3) to a spacestation at position (4,7,5) in one hour, both departing and arriving with 0 velocity, and arriving with 0 acceleration. You are able to fire the engines so that the accereration of the spaceship is at time t (measured in hours after the departure) is given by the formula : a(t) = t^2b + tc +d for any choice of the constant vectors b,c and d. Determine choices of these constant vectors so that you can complete your mission.

so far this is what i have:

a(0)= d
a(1)= b+c+d = 0

v(t)= (1/3)*t^3*b + (1/2)*t^2*c +d*t +k_1
v(0)=0+k_1 so k_1=0
v(1)= (1/3)*b + (1/2)*c + d = 0

r(t)= (1/12)*t^4*b + (1/6)*t^3*c + (1/2)*t^2*d +k_2
r(0)= 0+ k_2 so k_2 = (1,2,3)
r(1)= (1/12)*b + (1/6)*c + (1/2)*d +(1,2,3) = (4,7,5)

i then tried to solve them in a linear system using t=1, since i have 3 variables and 3 equations, but im not sure how i would handle the r(1) situation since i have two points in the equation, and the rest are vectors.

can anyone help me out here?


thanks

[Atheist]

actually you have
a(1) = b+c+d = (0,0,0)
v(1) = (1/3)b + (1/2)c +d = (0,0,0)
r(1) = (1/12)b + (1/6)c + (1/2)d + (1,2,3) = (4,7,5)

That´s because velocity and acceleration are also vectors, not scalars. You can either solve these equations for b,c,d like you´d do for real numbers (well, as long as you only add them or multiply them with real numbers, but that´s sufficient) or, if you feel uncomfortable doing that, break up these equation by b=(bx,by,bz), c=(cx,cy,cz), d=(dx,dy,dz) into a system of 9 equations with 9 unknowns (not recommended).

If your problem is that P0=(1,2,3) and P1=(4,7,5) are points in space while a,b,c are vectors and you are afraid of mixing points and vectors then remember that the difference P1-P0 = r(1)-r(0) = (4,7,5)-(1,2,3) between start- and endpoint is a vector.

Hope that helps.


Sidenote: Thumbs up that you presented your work on the problem - and even in a very readable form. That´s seen way too seldom when people ask for homework help.