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

## Main Question or Discussion Point

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

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

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Pengwuino
Gold Member

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

Exactly. The velocity of an object is simply the time-derivative of its position function.
Maybe I'm incredibly rusty on my calculus, but isn't the time-derivative of the first equation the following?

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

jtbell
Mentor

##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

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

##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

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

$$v_y(t) = v_{yi} + a_y t$$
Thank you!

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