for a constant acceleration "a", write an equation relating the final velocity vf and time tf to the initial velocity vi and time ti.
To write an equation relating the final velocity vf and time tf to the initial velocity vi and time ti for a constant acceleration a, we can use the kinematic equation:
vf = vi + a(t - ti)
In this equation, vf represents the final velocity, vi represents the initial velocity, a represents the constant acceleration, t represents the current time, and ti represents the initial time.
To derive this equation, we start with the kinematic equation for velocity:
vf = vi + at
This equation relates the final velocity vf to the initial velocity vi and acceleration a at a given time t. However, this equation assumes that acceleration is constant throughout the entire time interval. To adapt it to account for a different starting time ti, we modify the equation as follows:
vf = vi + a(t - ti)
By subtracting the initial time ti from the current time t, we can determine the time interval that the object has been accelerating from ti to t. Multiplying this time interval by the constant acceleration a and adding it to the initial velocity vi gives us the final velocity vf.
So, the equation vf = vi + a(t - ti) relates the final velocity vf and time tf to the initial velocity vi and time ti for a constant acceleration a.