Motion in Space: Velocity and Acceleration: Examples

1. Find the velocity and acceleration and force (mass = 5kg) for the following position function: <2t^2,-t,t^3>

Anwser:

For this anwser, we need to activate the VectorCalculus package:

>	with(VectorCalculus);

Now, take derivates for the velocity and acceleration (using diff command):

>	diff(<2*t^2,-t,t^3>,t);

>	diff(<2*t^2,-t,t^3>,t,t);

For the force, take the acceleration times the mass:

>	<4,0,6>*5;

2. Find the tangential and normal components of acceleration for the following position function: <t,t^2,t^2/2>

Anwser:

First, find the first and second derivatives of the function:

>	diff(<t,t^2,t^2/2>,t);

>	diff(<t,t^2,t^2/2>,t,t);

Now, we can take the dot product of the two components and put everything in the formula:

>	<1,2*t,t>.<0,2,1>;

>	(5*t)/sqrt((1)^2+(2*t)^2+(t)^2);

Next, for the normal component we use the derivatives from above, and take the cross product of r and r':

>	<1,2*t,t>&x<0,2,1>;

Then put the numbers in the equation:

>	sqrt((0)^2+(-1)^2+(2)^2)/sqrt((1)^2+(2*t)^2+(t)^2);

>