Kinematic Equations: Vector & Scalar Formulas

[fog37]

Hello Forum,
the kinematic equations for motion with constant acceleration are vector equations which can each be expanded into 3 scalar (or component equations). The vector equations are:

v_f = v_0 + a (Delta_t)

r_f = r_0 + v_0 (Delta_t) + (0.5) a (Delta_t)^2

r_f = r_0 + (0.5) (v_f + v_0) (Delta_t)

From these three vector equation we can write the corresponding scalar equations for the x,y and z components.

What about the scalar equation (v_f)^2 = (v_0)^2 +2a (x_f -x_0) ? It can be written for each scalar component.
What is the corresponding vector equation for it from which it comes from?

thanks
fog37

 

[jtbell]

It can be written for each scalar component.

That is, $$v_{xf}^2 = v_{x0}^2 + 2a_x (x_f - x_0) \\ v_{yf}^2 = v_{y0}^2 + 2a_y (y_f - y_0) \\ v_{zf}^2 = v_{z0}^2 + 2a_z (z_f - z_0)$$ Add the three equations together. Are you familiar with the vector dot product?

 

[fog37]

Thanks! I see how the addition of the three gives a single equation with dot products:

[ v_f dot v_f ] = [ v_0 dot v_0 ]+ 2 [a dot (r_f - r_0) ]

correct? Where does this equation come from? I guess it derives from that single differential equation dv/dt = a ...

 

[jtbell]

I think you can get it by combining two of the equations in your first post. Note which variable is "missing" from this equation.