differentiation in kinematics using equations
Answers
Calculus is an advanced math topic, but it makes deriving two of the three equations of motion much simpler. By definition, acceleration is the first derivative of velocity with respect to time. Take the operation in that definition and reverse it. Instead of differentiating velocity to find acceleration, integrate acceleration to find velocity. This gives us the velocity-time equation. If we assume acceleration is constant, we get the so-called first equation of motion [1].
a =
dv
dt
dv = a dt
v
⌠
⌡ dv
v0 =
t
⌠
⌡ a dt
0
v − v0 = at
v = v0 + at [1]
Again by definition, velocity is the first derivative of position with respect to time. Reverse the operation in the definition. Instead of differentiating position to find velocity, integrate velocity to find position. This gives us the position-time equation for constant acceleration, also known as the second equation of motion [2].
v =
ds
dt
ds = v dt
ds = (v0 + at) dt
s
⌠
⌡ ds
s0 =
t
⌠
⌡ (v0 + at) dt
0
s − s0 = v0t + ½at2
s = s0 + v0t + ½at2 [2]
↗Kinematics as Approach of Calculus
