In this problem, we're asked to calculate the initial speed an arrow is released with, given the horizontal distance traveled and the initial and final directions of motion.
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 a horizontally launched projectile, we also know that v0y = 0.
Finally, we might need to know how to decompose a velocity vector into its x- and y-components:
Or how to get the magnitude and angle of a velocity vector if we know the components:
Step 1. The problem gives us the horizontal distance the arrow traveled and the angle of its final velocity, as well as the fact that it's shot horizontally. So our known, unknown, and target variables are as follows:
In the x direction:
i. Δx = 61.0 m (known)
ii. v0x = ? (target)
iii. vfx = ? (unknown)
iv. t = ? (unknown)
v. ax = 0 m/s2 (known)