The position function from given velocity or acceleration function

[2x2lcallingcq]

Problem:

Find the position function from the given velocity or acceleration function.

a(t) = <e^-3t,t,sint>, v(0)=<4,-2,4>, r(0)=<0,4,-2>

Solution:

To find the answer the integral must be taken...

Integral of a(t) = <-(1/3)e^(-3t), (1/2)t^2, -cos(t)>

Since taking it with respect to t, it becomes velocity


(acceleration = x/t/t distance over time 2)

Integral of acceleration with respect to t is... (xt^-1, x/t)

Since that is just the velocity function, you do have an initial velocity at v(0)...(time at 0)
adding these to each part of the integral.

The velocity function is this: <-(1/3)e^(-3t)+4, (1/2)t^2-2, -cos(t)+4>

Again, the integral must be taken and it becomes the position function

x(t) = <(1/9)e^(-3t)+4t, (1/6)t^3-2t, 4t-sin(t)>

And add constants again r(0)

therefore the answer r(t) (x(t)) = <(1/9)e^(-3t)+4t+0, (1/6)t^3-2t+4, 4t-sin(t)-2>


BUT my teacher said the first and the third portions are wrong? I do not understand how.

 

[Mark44]

Problem:

Find the position function from the given velocity or acceleration function.

a(t) = <e^-3t,t,sint>, v(0)=<4,-2,4>, r(0)=<0,4,-2>

Solution:

To find the answer the integral must be taken...

Integral of a(t) = <-(1/3)e^(-3t), (1/2)t^2, -cos(t)>

Put the constants in right away.
v(t) = <-(1/3)e^-3t + c₁, (1/2)t² + c₂, -cos(t) + c₃>
You're given that v(0) = <4, -2, 4>, so you can solve for the three constants. It's not a simple matter of adding them in as you did.

Do the same thing when you find r(t).

Since taking it with respect to t, it becomes velocity


(acceleration = x/t/t distance over time 2)

Integral of acceleration with respect to t is... (xt^-1, x/t)

Since that is just the velocity function, you do have an initial velocity at v(0)...(time at 0)
adding these to each part of the integral.

The velocity function is this: <-(1/3)e^(-3t)+4, (1/2)t^2-2, -cos(t)+4>

Again, the integral must be taken and it becomes the position function

x(t) = <(1/9)e^(-3t)+4t, (1/6)t^3-2t, 4t-sin(t)>

And add constants again r(0)

therefore the answer r(t) (x(t)) = <(1/9)e^(-3t)+4t+0, (1/6)t^3-2t+4, 4t-sin(t)-2>


BUT my teacher said the first and the third portions are wrong? I do not understand how.

 

[2x2lcallingcq]

Put the constants in right away.
v(t) = <-(1/3)e^-3t + c₁, (1/2)t² + c₂, -cos(t) + c₃>
You're given that v(0) = <4, -2, 4>, so you can solve for the three constants. It's not a simple matter of adding them in as you did.

Do the same thing when you find r(t).

so I just put the constants in and then solve? I'm still confused.

 

[Mark44]

You are given that v(0) = <4, -2, 4>, and from your work you can substitute 0 for t to get v(0) = <-(1/3)e^-3*0 + c₁, (1/2)0² + c₂, -cos(0) + c₃>.

From this you can solve for the constants.