A hoodlum throws a stone vertically downward with an initial speed of 12.0m/s from the roof of a building, 30.0m above the ground. (a) How long does it take the stone to reach the ground? (b) What is the speed of the stone at impact?

We're asked for the *time of flight* and *final velocity* of the vertically-thrown stone.

Whenever we have a problem where an object is dropped or thrown from some height, it is affected only by the constant acceleration of gravity (*g* = 9.8 m/s) after being released, and therefore we can use the kinematics (UAM) equations. Because the motion is in the vertical direction, we'll replace Δ*x* with Δ*y*. Our standard coordinate system has the +*y*-axis pointing upwards, so we say *a *= −*g* = −9.8 m/s^{2}. To solve, we'll follow these steps:

- Identify the
for each part of the problem.__target variable__ - Identify
__which variables are__—remember that__known__*only*(Δ**3**of the**5**variables*y*,*v*_{0},*v*,_{f}*a*, and*t*)*are needed*to solve any kinematics problem, and we already know what*a*is. __Choose a UAM equation__with**only one unknown**, which should be our**target variable**.__Solve the equation__for the target variable, then__substitute known values__and__calculate__the answer.

$\overline{)\mathbf{}{{\mathit{v}}}_{{\mathit{f}}}{\mathbf{}}{\mathbf{=}}{\mathbf{}}{{\mathit{v}}}_{{\mathbf{0}}}{\mathbf{}}{\mathbf{-}}{\mathit{g}}{\mathit{t}}\phantom{\rule{0ex}{0ex}}{\mathbf{\u2206}}{\mathit{y}}{\mathbf{=}}{\mathbf{}}\mathbf{\left(}\frac{{\mathit{v}}_{\mathit{f}}\mathbf{+}{\mathit{v}}_{\mathbf{0}}}{\mathbf{2}}\mathbf{\right)}{\mathit{t}}\phantom{\rule{0ex}{0ex}}{\mathbf{\u2206}}{\mathit{y}}{\mathbf{=}}{\mathbf{}}{{\mathit{v}}}_{{\mathbf{0}}}{\mathit{t}}{\mathbf{-}}{\frac{1}{2}}{\mathit{g}}{{\mathit{t}}}^{{\mathbf{2}}}\phantom{\rule{0ex}{0ex}}{\mathbf{}}{{{\mathit{v}}}_{{\mathit{f}}}}^{{\mathbf{2}}}{\mathbf{=}}{\mathbf{}}{{{\mathit{v}}}_{{\mathbf{0}}}}^{{\mathbf{2}}}{\mathbf{}}{\mathbf{-}}{\mathbf{2}}{\mathit{g}}{\mathbf{\u2206}}{\mathit{y}}}$

Remember that **down** is the *−y*-direction, and __make sure you're using the right sign__ for any velocities and displacements in the problem!

Vertical Motion & Free Fall

Vertical Motion & Free Fall

Vertical Motion & Free Fall

Vertical Motion & Free Fall