Question

The acceleration due to air resistanceacceleration due to air resistance of a particle movingof a particle moving along a straight linealong a straight line at time t is proportional to the secondsecond power of its velocity vvelocity v. The differentialâ equation, with proportionality constantâ k, is_______.

Answer 1

`a'(t) = 2k*v'(t)*v(t)`

Explanation:

According to the data provided, the acceleration can be modeled by the following equation:

`a(t) = kv(t)^2`

Where a(t) is the acceleration as a function of time, and v(t) is the velocity ad a function of time.

Applying the chain rule, the differential equation, with proportionality constant k, is:

`(d(a(t)))/(dt)=(d(kv(t)*v(t)))/(dt) \\a'(t) = k*(v(t)*v'(t)+v'(t)*v(t))\\a'(t) = 2k*v'(t)*v(t)`