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Hi.
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In this video, we're given that there's a projectile which is fired at 170 meters per second at an angle of 60 degrees, starting from 100 meters above the ground.
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And we're asked to find three things.
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The range, so like the distance that the projectile travels, the max height, so the highest point that the projectile reaches, and the speed at impact.
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So to do all of these problems, what we're going to want to do first is come up with parametric equations describing the x and y position of the projectile.
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So since we're given the firing speed, we're going to start with the velocity.
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So the projectile is launched at 60 degrees with 170 meters per second speed.
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So what we want to do first is find the initial y velocity and the initial x velocity.
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So this, as you might have seen before, is a simple matter of trigonometry.
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So basically what we're doing is we're finding the base and height of this right triangle here.
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So using the sine and cosine functions, we see that the y is equal to, let's see, this is 170 times sine.
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Of 60 degrees, which is 170 root 3 over 2.
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And then vx is 170 times cosine of 60 degrees, which is equal to 170 over 2.
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So these are our initial velocities.
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So we can come up with, um, with the velocity equations now.
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The velocity y at t is going to be, um, i should start with, or sorry, no, actually, let me, let me start with, um, we should really start with the acceleration.
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Um, so in the, in the x direction, we have, uh, we have no acceleration and we have, um, initial velocity given here.
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So we can write that the x position at times t is just going to be 170 over two times t.
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Um, so the x is simplest because there's no acceleration.
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Um, but then with y of t, right? it's going to be of the form a t squared plus initial velocity times t, plus a constant c.
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Oh, and also in x, we're assuming that it starts at x equals zero.
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But we know the acceleration is negative 9 .8.
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We know the velocity is 170 root 3, 2 times t.
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And then we're also given that the initial height of the projectile is 100.
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So, we have the two parametric equations describing the path of the of the projectile.
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So now i think we're ready to solve a through c.
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So for a, we want to find the range of the projectile.
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So what that means is how far in the x direction does the projectile land? so the first thing we want to do is we want to, because the parametric equations are in terms of time, we want to figure out how long it takes for the projectile to to fall to the ground.
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So in other words, what we want to do is solve y of t equal to zero.
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And of course, we're going to get two solutions because y of t is a parabola, but one of the solutions will be negative.
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So if you solve this using the quadratic formula or a calculator, however you want, we get that it takes 15 .674 seconds to land.
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So this is the landing time.
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So then if we want to find the range, this is just equal.
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To the x value at 15 .674...