Distance to accelerate to velocity (vector division???)

Call Me Jimmy

I've only just started studying physics as I'm trying to simulate a scenario on the computer.

I have an equation for the distance it will take to accelerate from an initial velocity to a final velocity at a constant rate.

d = (vf * vf) - (vi * vi) / 2a

I tested it in one dimension (where the vi, vf and a are scalars (I know technically they never are) and it worked correctly.

What I don't understand is in moving this to 3 dimensions where vf, vi and a are vectors, how is this computed? I've read that vector division is undefined and it's not working out.

I am definitely misunderstanding something. How would the operation of dividing the difference between the velocity vectors divided by 2 times the acceleration vector happen? Should I be computing the length of the vectors which in turn will give scalars? That doesn't seem like it will be accurate.

I've gone googling but am getting more confused.

Thanks.

Morbert

Call Me Jimmy wrote: »

I've only just started studying physics as I'm trying to simulate a scenario on the computer.

I have an equation for the distance it will take to accelerate from an initial velocity to a final velocity at a constant rate.

d = (vf * vf) - (vi * vi) / 2a

I tested it in one dimension (where the vi, vf and a are scalars (I know technically they never are) and it worked correctly.

What I don't understand is in moving this to 3 dimensions where vf, vi and a are vectors, how is this computed? I've read that vector division is undefined and it's not working out.

I am definitely misunderstanding something. How would the operation of dividing the difference between the velocity vectors divided by 2 times the acceleration vector happen? Should I be computing the length of the vectors which in turn will give scalars? That doesn't seem like it will be accurate.

I've gone googling but am getting more confused.

Thanks.

The equation would be applied to each component of the vectors individually.

For example, using cartesian coordinates (x,y,z) let's say your vectors are

vi = (0,0,0)
vf = (1,2,3)
a = (.2,.4,.6)

Then your distance would be

d_x = (1*1-0*0)/(2*.2) = 1/.4 = 2.5

d_y = (2*2-0*0)/(2*.4) = 4/.8 = 5

d_z = (3*3-0*0)/(2*.6) = 9/1.2 = 7.5

d = (2.5, 5, 7.5)

You have to be a little careful about what you set your acceleration to be. To check if your velocities and acceleration are compatible, you should check that

(vf_x-vi_x)/a_x = (vf_y-vi_y)/a_y = (vf_z-vi_z)/a_z

Call Me Jimmy

Really? Thanks so much!

I take it then that applies for all such equations?

Is that something one would learn as part of physics, that when dealing with vector quantities in equations, that each equation should be applied to each component?

It's just in the bit of theory I have read (honestly not much) and from searching I couldn't find that explanation.

Much appreciated!

On the last bit about compatibility, do you mean in the context of this equation? Does that equation read as 'the distance for x component' must equal the distance of the other two? That doesn't make sense so I must be misunderstanding.

Call Me Jimmy

Also the velocity is derived (the right word?) from the acceleration so it should be compatible, yes?

Morbert

Call Me Jimmy wrote: »

Really? Thanks so much!

I take it then that applies for all such equations?

Is that something one would learn as part of physics, that when dealing with vector quantities in equations, that each equation should be applied to each component?

It's just in the bit of theory I have read (honestly not much) and from searching I couldn't find that explanation.

It works in the case of kinematics because of the way these equations are derived. I.e. The equation does not have vectors. Rather, the equation has scalar components of vectors. And so if you write your vectors in terms of three perpendicular vectors x,y and z (called basis vectors), then your vectors will have three components each, and the equation must be applied to these components.

Equations that actually involve vector quantities can look quite different, as they can have vector operators like the "dot product" or "cross product".

On the last bit about compatibility, do you mean in the context of this equation? Does that equation read as 'the distance for x component' must equal the distance of the other two? That doesn't make sense so I must be misunderstanding.

It reads "the time for the x component to go from vi_x to vf_x must be equal to the time for the other components to do the same".

For example, it is impossible for an object with acceleration (1,2,3) to go from an initial velocity (0,0,0) to a final velocity (1,1,1)