Definition
The path a projectile travels in motion.
Yes, that is really a person being shot out of a cannon.
A Few Comments Before We Start
There is going to be a lot of math in this section. Do not freak out. The math will not hurt you. Most of this is very much common sense oriented. Don’t think of it as math; think about how you threw things last week with Levers. The numbers just stand behind what you already know in your head.
CAN HAVE FUN WITH PROJECTILES!
How it works
As the projectile moves its velocities push it both up and across. It will always have both velocities for this reason.
When the projectile goes up, it fights the force of gravity until its vertical velocity is zero and it stops going up.
The vertical velocity is zero only for a second when it reaches the maximum height. Then the projectile is falling and accelerating with gravity.
The entire time the vertical velocity of the projectile is changing, the horizontal velocity is staying the same. The projectile is always moving horizontally. For this reason it always has the same horizontal velocity. At every moment the projectile is moving, both velocities are always acting upon it. Whether it goes up and across or down and across it is doing two things at once, meaning two velocities. Even when one velocity is zero it is acting upon it, because the projectile isn’t moving!
The Velocities
Since the horizontal and vertical velocities control the path of the projectile they are what you need to figure out for trajectory. If we know how the projectile is being pushed, we know where it is going to go.
The velocities of a projectile are a bit trickier. Since the projectile is symmetrical, we only have to deal with half of it to find both velocities. The horizontal velocity is always the same and the vertical velocity is effected by gravity (which we know) after the half way point.
Getting Crazy!
THE GREAT SOH CAH TOA
Write this down! It is UBER important!
.....
No really. Write it down.
Copy this on a piece of paper.
Soh- sin = opposite over hypotenuse.
Cah- Cos = Adjacent over total velocity
Toa- Tan = opposite over adjacent.
Your opposite and adjacent sides, as well as your hypotenuse could also be called vertical velocity, horizontal velocity, and total velocity.
For more information on this assingment, see Mr. Evans.
How much time?
(Final velocity - vertical velocity)/acceleration
Don’t freak out. This is NOT as hardcore as it looks. You already know all the numbers.
Acceleration means a CHANGE IN velocity. When the projectile is going up, it is slowing down. It is changing its velocity due to gravity pulling on it. It is slowing down at the acceleration of gravity.
You know the acceleration of gravity = 9.8 meters/per second2
Remember, you are slowing down so the value of acceleration is NEGATIVE.
To figure out how high the projectile will go to reach its maximum height use this formula:
(Vertical velocity x time) + (.5 x acceleration x time2 )
Again. Don’t freak out. This is NOT as hardcore as it looks. You already know all the numbers.
Let’s Just Think About It….
8.34 - 4.17 = 4.17
You know the vertical velocity. You already figured it out with SOH, CAH, TOA. Remember, we fired the projectile at 65 degrees. Just fill the number in.
Time refers to the specific moment in air you want to find the height of. You just figured that out. At what time is the projectile at its maximum height?
.92 seconds
How Far Will It Go?
Distance = Horizontal velocity x time
By now, this one should look easy.
Let’s Just Think About It…
Distance = 7.76 meters
For time, you need to figure out what you want to know.
You already know the horizontal velocity. You figured it out earlier with SOH, CAH, TOA. The projectile was fired at a 65 degree angle so the horizontal velocity is 4.22 meters per second.
How much distance does the projectile travel before it hits the ground?
The trajectory is symmetrical. If it reaches its maximum height half way through the total flight, multiply the max. height time by 2 and you get the total flight time.
92 x 2 = 1.84
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