Youve learned the basic mechanics equations that describe motion in one dimension, but now its time to take it one step further with motion in two dimensions. There are a bunch of concepts here that you should take to heart regarding a projectile moving in multiple dimensions. One big one to take away is the fact that you can use variables in the y-direction to solve for unknowns in the x-direction. As always, understanding the reasons behind the math is just as important as the math, and this section is a great feet wetter into that fact. These problems are all taken from a stationary standpoint, which will change in the next section.
Projectiles are objects acted upon only by gravity.
Projectiles launched at angles move in parabolic arcs.
Vertical motion and horizontal motion are completely independent. Horizontal motion does not affect vertical motion, and vice versa.
An object will travel the maximum horizontal distance across level ground with a launch angle of 45 degrees.
Lecture Slides are screen-captured images of important points in the lecture. Students can download and print out these lecture slide images to do practice problems as well as take notes while watching the lecture.
Hi everybody and welcome back to Educator.com.0000
In this lesson we are going to talk about projectiles.0003
Our objectives are going to be to sketch the theoretical path of a projectile, to recognize the independence of the vertical and horizontal motions of a projectile, and to solve problems involving projectile motion for projectiles fired horizontally and projectiles fired at an angle.0006
A projectile is an object that is acted upon only by gravity.0029
In reality, air resistance plays a role -- we all know it does, but for the purposes of this analysis we are going to neglect air resistance.0034
Typically, projectiles are objects launched at an angle.0043
So a projectile is probably something that gets launched up in the air, comes back down or is launched, comes back down horizontally, or is launched down and travels that direction -- typical projectile.0046
Let us start by talking about the path of the projectile.0062
Projectiles launched at an angle move in parabolic arcs.0065
It is coming back down to the same height it was launched from -- it will travel up in an arc and back down -- split it in two and you should have a mirror image of each other.0099
Now the distance it travels along the ground is known as its horizontal range.0108
We could also look at the max height that the projectile reaches.0116
And one key, if you want a projectile launched from level ground to go the furthest possible distance, the angle you want to launch it at -- 45 degrees. Okay?0123
Let us take a look at the independence of motion.0136
Projectiles, objects launched in two dimensions -- it is the same physics we have been doing. They are easy.0140
The same thing we have been doing with kinematic equations, we are just going to do it again.0148
The only difference is we are going to treat the vertical aspect of the projectile and the horizontal aspect separately.0151
As long as we keep them separate, we can keep doing all the same math we have been doing.0160
No real new analysis needs to be brought into the fold.0165
Vertical is just like the free fall problems -- and horizontal -- even simpler. There is no acceleration.0169
Gravity never pulls you sideways -- does not happen. Gravity only pulls down.0174
So as long as we treat them separately our analyses get pretty easy.0180
What we are going to have to remember though, is breaking vectors up into components.0184
If for example, we have an initial velocity for a vector -- some initial velocity launched at some angle theta -- we need to simplify it.0188
We need to make this easy by breaking it up into an X component -- V initial x -- which hopefully you remember is V initial cos theta and V initial y component -- V initial y, which is V initial sin of the angle theta.0199
Then we are going to deal with this V initial x and this V initial y vector separately.0220
All right, let us make our table for what we know vertically.0307
Vertically, our object is going to go down first so let us call down the positive y direction.0311
We have V initial y, V final y, delta y, Ay and T.0317
V initial y for an object launched horizontally, if all of its velocity is this way, there is no initial vertical component. That is 0.0326
We do not know what its final vertical component is going to be but we know its displacement vertically is going to be 2 m because that is how far it travels vertically before it hits the ground.0334
The acceleration in the y direction if we call down positive is 10 m/s2.0344
The only thing that is the same between these two tables is the time horizontally and the time vertically -- regardless it hits the ground at the same amount of time.0351
Whatever time we have time vertically must be the time horizontally.0360
So let us solve for the time with these things then we can solve for Δx, which is what we are after -- how far the ball travelled horizontally before reaching the ground.0364
Well to do this, if I want to find time -- Oh! We have this nice V initial y = 0.0376
So let us use delta y = V initial y x T + 1/2A yT2, and again our trick, V initial y is 0 so that whole term goes away.0382
Now then we can rearrange this to solve for t.0395
T is going to be equal to delta y/Ay -- that is T2, so we are going to need to take the square root.0399
So that is 2 x 2 m/10m/s2 square root, or 0.63 seconds.0410
If the ball is in the air 0.63 seconds vertically, it is in the air 0.63 seconds horizontally.0420
Now to find the horizontal displacement, this delta x -- delta x = Vx times t. V times t. 0427
That is 42 m/s times 0.63 seconds or 26.5 meters.0439
How far did the ball travel horizontally? 26.5 meters.0451
Let us take a look at another example problem.0460
We have a diagram here showing the path of a stunt car driven off of a cliff.0462
If we neglect friction, let us compare the horizontal component of the car's velocity at point A., where it has some horizontal component of the velocity to the horizontal component of the car's velocity at B.0467
Well the trick here is realizing that horizontally, there is no acceleration.0480
Nothing is causing the car to speed up or slow down horizontally because we are neglecting friction.0485
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