Spaceship Mission: Vector Functions for Flight from (1,2,3) to (4,7,5) in 1 Hour

[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 here's 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 I am 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.

 

[wais]

for reaching out for help! It looks like you have made good progress so far in setting up the equations for the vector functions. To solve for the constant vectors b, c, and d, you can use the following steps:

1. Use the equation a(t) = t^2b + tc + d and the given information that the acceleration is 0 at both the departure and arrival times (t=0 and t=1) to set up a system of equations. This will give you two equations: a(0)=0 and a(1)=0.

2. Use the equation v(t) = (1/3)*t^3b + (1/2)*t^2c + dt + k_1 and the given information that the velocity is 0 at the departure time (t=0) to set up a third equation: v(0)=0.

3. Use the equation r(t) = (1/12)*t^4b + (1/6)*t^3c + (1/2)*t^2d + k_2 and the given information that the position is (1,2,3) at the departure time (t=0) to set up a fourth equation: r(0)=(1,2,3).

4. You now have four equations with four unknowns (b, c, d, and k_1). You can use any method (such as substitution or elimination) to solve for these unknowns. Once you have found the values for b, c, and d, you can plug them into the equations for v(t) and r(t) to find the values for k_1 and k_2, respectively.

5. Finally, check that the values you have found satisfy the equation r(1) = (1/12)*b + (1/6)*c + (1/2)*d + k_2 = (4,7,5). If they do, then you have successfully found the constant vectors that will allow you to complete your mission. If not, then you may need to check your calculations or try a different approach.

I hope this helps and good luck with your mission!