# Is there some sort of calculus relationship between these two kinematics equations?

1. ### tahayassen

273
$${ y }_{ f }={ y }_{ i }+{ v }_{ yi }t+\frac { 1 }{ 2 } { a }_{ y }{ t }^{ 2 }\\ { v }_{ yf }={ v }_{ yi }+{ a }_{ y }t$$

It almost looks like the second equation is the derivative of the first equation with respect to time.

2. ### Pengwuino

7,118
Re: Is there some sort of calculus relationship between these two kinematics equation

Exactly. The velocity of an object is simply the time-derivative of its position function.

3. ### tahayassen

273
Re: Is there some sort of calculus relationship between these two kinematics equation

Maybe I'm incredibly rusty on my calculus, but isn't the time-derivative of the first equation the following?

$$0={ v }_{ yi }+{ a }_{ y }t$$

### Staff: Mentor

Re: Is there some sort of calculus relationship between these two kinematics equation

##v_{yf}## isn't a constant, it's a variable, more specifically the dependent variable, a function of t. Written as functions, your two equations are

$$y(t) = y_i + v_{yi} t + \frac{1}{2}a_y t^2 \\ v_y(t) = v_{yi} + a_y t$$

5. ### tahayassen

273
Re: Is there some sort of calculus relationship between these two kinematics equation

Thank you!

6. ### tahayassen

273
Re: Is there some sort of calculus relationship between these two kinematics equation

Any way I can derive the below equation from the position and velocity equations?

$${ { v }_{ yf } }^{ 2 }={ { v }_{ yi } }^{ 2 }+2{ a }_{ y }({ y }_{ f }-{ y }_{ i })$$