An object is launched into the air at an angle of 60 degress with respect to the horizontal. What are the velocities in the x and y directions? When studying kinematic motion, projectile motion, throwing objects up in the air, shooting them from cannons, flinging them from trebuchets, you will constantly be concerned with the velocity in the horizontal or x-direction and the velocity in the vertical or y-direction. This is mostly due to the inclusion of gravity. Since gravity works roughly straight up and down, The motions in the vertical and horizontal directions can be considered independent, and then combined together as a final answer. Usually, the object being launched is not straight up and down. Thus, we’ll have to take this velocity vector, and find out how much of it is directed in the vertical direction and how much in the horizontal direction. We can do this with ease by implementing the 5 steps from last episode.
Step 1: Identify the coordinate axis and draw it on the paper. This is critically important. Usually this step is as simple as drawing the y-direction vertically and the x-direction horizontally. However, there are sometimes cases where it may be easier to make the coordinate axis in some other orientation. So it is very important that everyone understands the coordinate system as the very first item.
Step 2: Identify the vector and draw it on the paper. In this case we have a velocity vector v which is 60 degrees with respect to the horizontal, which has been conveniently aligned with our x-axis. The y-axis, aligned vertically, will work directly opposite gravity. This will be very convenient in problems in force problems and when we dive deeper into kinematics.
Step 3: Identify the angle. In this case it is very straightforward: 60 degrees with respect to the horizontal. Sometimes there will be some more calculations required. Usually the “with respect to the horizontal” is the easiest angle to work with. I pray for this angle to be given in problems. Writing it down on the drawing is critical. It leaves no room for confusion and makes your work complete and precise. It is wise to only place one vector per diagram because with the vectors and angles and labels things can get crowded and messy.
Step 4: Use sine and cosine relationships for component definition. For this very standard case: x-direction aligned with the horizontal, y-direction pointed vertical, and the angle given w/r/t horizontal, you will always be able to use cosine for x and sine for y. This is because everything is set up just as in our unit circle and the coordinates for our radius as we move around the circle is [cos,sin].
Step 5: Box your answer, you’re done! You can verify these calculations in your head if you use the sines and cosines by counting to 4. The final answer is wonderfully generalized so that any vector v at this angle can be plugged in. This is the best type of answer: correct, robust, and specific.
