When an object is thrown up vertically, the time taken to reach the highest point and the time taken to fall back to the initial projection can be compared by constructing a mathematical model using a differential equation to describe the phenomenom.
An object of mass m kg is projected vertically upward from the ground with initial velocity u ms-1. Assume that the forces acting on the object are the gravitational force and the retarding forces due to air resisitance with a magnitude of α│v│, where α is a positive constant and v is the velocity of the object at time t.

m dv/dt=-∝v-mg

(a)Express v in term of t.
(b)Find the height s in term of t.
(c) If the mass of the object is 0.1 kg, the initial velocity is 5 ms-1 , α is 1.0 kgms-1 and g is 10ms-2.
i)Calculate the time taken for the object to reach the highest point
ii)Estimate the time taken for the object to descend from the highest point to the ground.

thanks!

$m \frac{dv}{dt}= -\alpha v -mg$
$m\frac{dv}{-\alpha v - mg}= dt$