I have this problem:

A student throws a set of keys vertically upward to her sorority

sister, who is in a window 3.30 m above. The second student catches

the keys 1.60 s later1) With what initial velocity were the keys thrown?

2) What was the velocity of the keys just before they were caught?

For question #1 I got 10 m/s, which is correct. However, for the second question I also got the right answer but at first I did not know what formula to use since there are two kinematic equations specific for final Velocity

(1) $v = v_o + at $

(2) $v^2 = {v_o}^2 + 2ad $

I know the correct result comes from using formula (1) but I do not understand why using both formulas I get different results. How then would I know which formula to use in different situations?

## Best Answer

I'm not sure why you're using formula (1) as it has two unknowns $v$ and $v_0$; the final and initial velocities respectively.

Assuming you took gravitational acceleration $a=-g\approx -10 \mathrm{ms^{-2}}$ then I would use $$d=v_0 t - \frac12 g t^2$$ which when solving for $v_0$ gives $$v_0 =\frac{3.3+0.5\cdot 10 \cdot 1.6^2}{1.6}=10.0625\approx 10\mathrm{ms^{-1}}$$ for the intitial velocity.

Then using formula (2) we have

$$v=\sqrt{10.0625^2-2\cdot 10 \cdot 3.3}=5.9375\mathrm{ms^{-1}}$$ for the final velocity.