This problem is asking us to find the **minimum initial velocity** of a projectile given the **direction** of the initial velocity and the **horizontal and vertical distance** it has to travel.

For **projectile motion problems in general**, we'll follow these steps to solve:

- Identify the
and__target variable__for each direction—remember that__known variables__*only*(Δ**3**of the**5**variables*x*or Δ*y*,*v*_{0},*v*,_{f}*a*, and*t*)*are needed*for each direction. Also, it always helps to sketch out the problem and label all your known information! __Choose a UAM__—sometimes you'll be able to go directly for the target variable, sometimes another step will be needed in between.**equation**for the target (or intermediate) variable, then**Solve**the equation__substitute known values__and__calculate__the answer.

The four UAM (kinematics) equations are:

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

We define our coordinate system so that the **+ y-axis is pointing upwards** and the

For projectiles with a** positive launch angle**, we __also__ need to know how to decompose a velocity vector into its *x*- and *y*-components:

$\overline{)\begin{array}{rcl}{\mathit{v}}_{\mathbf{0}\mathit{x}}& {\mathbf{=}}& \mathbf{\left|}{\stackrel{\mathbf{\rightharpoonup}}{\mathit{v}}}_{\mathbf{0}}\mathbf{\right|}\mathbf{}\mathbf{cos}\mathbf{}\mathit{\theta}\\ {\mathit{v}}_{\mathbf{0}\mathit{y}}& {\mathbf{=}}& \mathbf{\left|}{\stackrel{\mathbf{\rightharpoonup}}{\mathit{v}}}_{\mathbf{0}}\mathbf{\right|}\mathbf{}\mathbf{sin}\mathbf{}\mathit{\theta}\end{array}}$

A cannon, located 60.0 m from the base of a vertical 25.0-m-tall cliff, shoots a 15-kg shell at 43.0 above the horizontal toward the cliff. What must the minimum muzzle velocity be for the shell to hit the top edge of the cliff?

Frequently Asked Questions

What scientific concept do you need to know in order to solve this problem?

Our tutors have indicated that to solve this problem you will need to apply the Projectile Motion: Positive Launch concept. If you need more Projectile Motion: Positive Launch practice, you can also practice Projectile Motion: Positive Launch practice problems.

What professor is this problem relevant for?

Based on our data, we think this problem is relevant for Professor Hatch's class at UMASS.