The acceleration of a badminton birdie falling under the influence of gravity and a resistive force, such as air resistance, is given by a = dv/dt = g - bv, where g is the free-fall acceleration due to gravity and b is a constant that depends on the mass and shape of the birdie and on the properties of the medium. Suppose the birdie begins with zero velocity at time t = 0.

(a) Discuss qualitatively how the speed v varies with time from your knowledge of the rate of change dv/dt given by this equation. What is the velocity when the acceleration is zero? This is called the terminal velocity.

(b) Sketch the solution v(t) versus t without solving the equation. This can be done as follows: at t = 0, v is zero and the slope is g. Sketch a straight-line segment, neglecting any change in slope for a short time interval. At the end of the interval, the velocity is not zero, so the slope is less than g. Sketch another straight-line segment with a smaller slope. Continue until the slope is zero and the velocity equals the terminal velocity.