When an object is thrown vertically upward, it is under the effect of gravity and air resistance.For small objects, the force due to air resistance is numerically equal to some constant k times v, where v is the velocity of the object (in m/s) at timet (s).
This situation can be modelled as the differential equation shown below.
`m (dv)/(dt) = -F_R - m_g`
where,
m is the mass of the object in kg.
`(dv)/(dt)` is the acceleration of the object in `m/s^2`.
`F_R` is the force due to air resistance.
g is the acceleration due to gravity `(10 m/s^2)`.
A tennis ball of mass `0.050` kg is hit upwards with a velocity of 10m/s. An air resistance numerically equal to `0.4v` acts on the ball.
(i) Model the above situation using a differential equation.
(ii) Write an expression for the velocity of the ball in terms of the time.
Show your work.