This problem is asking us to find the minimum initial velocity of the cannonball 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 target variable and known variables for each direction—remember that only 3 of the 5 variables (Δx or Δy, v0, vf, 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 equation—sometimes you'll be able to go directly for the target variable, sometimes another step will be needed in between.
- Solve the equation for the target (or intermediate) variable, then substitute known values and calculate the answer.
The four UAM (kinematics) equations are:
We define our coordinate system so that the +y-axis is pointing upwards and the +x-direction is horizontal along the launch direction. That means ay = −g, and ax = 0 (because the only acceleration acting on a projectile once it's launched is gravity.)
For projectiles with a positive launch angle, we also need to know how to decompose a velocity vector into its x- and y-components: