Question

A ball is dropped off a high cliff, and 2 s later another ball is thrown vertically downward 

with an initial speed of 30 m/s. How long will it take the second ball to overtake the first?

Answer 1

Let's first write the vertical displacement of the first and second balls:

$$y_1=\dfrac{1}{2}gt^2,$$ $$y_2=v_0(t-2)+\dfrac{1}{2}g(t-2)^2.$$

When the second ball overtakes the first one $y_1=y_2$ and we can write:

$$\dfrac{1}{2}gt^2=v_0(t-2)+\dfrac{1}{2}g(t-2)^2,$$ $$\dfrac{1}{2}gt^2=v_0t-2v_0+\dfrac{1}{2}gt^2-2gt+2g,$$ $$30t-60-19.6t+19.6=0,$$ $$10.4t-40.4=0,$$ $$t=\dfrac{40.4}{10.4}=3.88\ s.$$