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# Projectile at an angle

## Video transcript

so I've got a rocket here and this rocket is going to launch a projectile maybe it's a rock of some kind with the velocity of 10 meters per second and the direction of that velocity is going to be 30 degrees 30 degrees upwards from the horizontal or the angle between the direction of the launch and the horizontal is 30 degrees and what we want to figure out in this video is how far does the rock travel we want to figure out how how far does it travel does it travel and to simplify this problem what we're going to do is we're going to break down this velocity vector into its vertical and horizontal components we're going to use a vertical component so let me just draw it visually so this velocity vector can be broken down into its vertical and it is and its horizontal components and its horizontal components so we're going to get some vertical component some amount of velocity in the upwards direction and we can figure we can use that to figure out how long will this rock stay in the air because it doesn't matter what its horizontal component is its vertical component is going to determine how quickly it decelerates due to gravity and then re accelerates and essentially how long it's going to be in the air and once we figure out how long it's in the air we can multiply it by we can multiply it by the horizontal component of the velocity and that will tell us how far it travels and once again the assumption that we're making in this video is that air resistance is negligible obviously if there was significant air resistance this horizontal velocity would not stay constant while it's traveling through the air but we're going to assume that it does that it does not change that it is negligible we could assume that we're doing this experiment on the moon if we wanted to have a if we want to view it in pure terms but let's solve the problem so the first thing we want to do is we want to break down this velocity vector we want to break down this velocity vector that has a magnitude of 10 meters per second and has an angle of 30 degrees with the horizontal we want to break it down into its X&Y components or its horizontal and vertical components so that's it's horizontal we draw a little bit better that's its horizontal component and then its vertical component looks like this this is its vertical component so let's do the vertical component first so how do we figure out the vertical component given that we know the hypotenuse of this right triangle and we know this angle right over here and the angle and the side this vertical component or the length of that vertical component or the magnitude of it is opposite the angle so we want to figure out the opposite we have the hypotenuse so once again we write down so so katoa sine is opposite over hypotenuse so we know that the sine the sine of 30 degrees the sine of 30 degrees is going to be equal to the magnitude of our vertical component so it's this is the magnitude of velocity I'll say velocity in the Y direction that's the vertical direction Y is the upwards direction is equal to the magnitude of our velocity of the velocity in the Y Direction divided by the magnitude of the hypotenuse or the magnitude of our original vector divided by 10 meters per second 10 meters per second and then to solve for this quantity right over here we multiply both sides by 10 and you get 10 sine of 30 10 sine of 30 degrees 10 sine of 30 degrees is going to be equal to the magnitude of our the magnitude of our vertical component and so what is the sine of 30 degrees and this you might have memorized this from your basic trigonometry class you could get the calculator out if you want but sine of 30 degrees is pretty straightforward it is 1/2 so sine of 30 degrees use a calculator if you don't remember that or you remember it now so sine of 30 degrees is 1/2 and so 10 times 1/2 is going to be 5 so and I forgot the unit's there so it's 5 meters per second is equal to the magnitude is equal to the magnitude of our vertical component get that in the right color it's equal to the magnitude of our vertical component so what does that do what we this this projectile because it's vertical component is five meters per second it will stay in the air the same amount of time as anything that has a vertical of a component of five meters per second if you threw a rock or projectile straight up with out of at a velocity five meters per second that rock or projectile will stay up in the air as long as this one here because they have the same vertical component so let's think about how long it will stay in the air since we're dealing with the situation where we're starting at the ground and we're also finishing at the same elevation and we're assuming that air resistance is negligible we can do a little bit of a simplification here although I'll do another version where we do it in the more complicated but I guess the way that applies to more situations we could say we could say well what is our change in velocity here so if we think about just the vertical velocity our initial velocity let me write it this way our initial velocity and we're talking let me label all of this so we're talking only in the vertical let me do all of the vertical stuff amudha in blue so vertical we're dealing with the vertical here so our initial velocity in the vertical direction our initial velocity in the vertical direction is going to be five meters per second it's going to be five meters per second and we're going to use the convention that up that up is positive and that down is negative and now what is going to be our final velocity we're going to be going up and we're going to be decelerated by gravity we're going to be stationary at some point and then we're going to start accelerating back down and if we assume that air resistance is negligible when we get back to ground level we will have the same magnitude of velocity but we'll be going in the opposite direction so our final velocity remember we're just talking about the vertical component right or right now we haven't even thought about the horizontal we're just trying to figure out how long does this thing stay in the air so it's final velocity is going to be negative five negative five meters per second if this is the initial velocity the final velocity is going to be looking like that same magnitude just in the opposite direction so what's our change in velocity in the vertical direction change in velocity in the vertical direction or in the Y direction is going to be our final velocity negative five meters per second minus our initial velocity minus five meters per second which is equal to negative 10 meters per second so how do we use this information to figure out how long it's in the air well we know we know that our vertical our change our change in our in our vertical velocity is going to be the same thing or it's equal to our acceleration in the vertical Direction times the change in time times the amount of time that passes by what's our acceleration in the vertical direction well it's the acceleration due to gravity or the acceleration that gravity then the force of gravity has on an object in freefall and so this right here is going to be negative 9.8 meters per second squared so this quantity over here is negative 10 meters per second we figure that out that's going to be the change in velocity negative 10 meters per second is going to be equal to negative 9.8 negative 9.8 meters per second squared times our change in time so to figure out the total amount of time that we are in the air we just divide both sides by negative 9.8 meters per second squared so we get let's just do that I want to do that in that same color so I do it in that's not well it's close enough so we get negative 9.8 meters per second squared negative 9.8 meters per second squared that cancels out and I get my change in time and I'll just get the calculator I have a negative divided by negative so that's a positive which is good because we want to go in positive time we assume that the elapsed time is a positive one and so what do we get if I get my calculator out I get my calculator out I have this is the same thing as positive ten divided by nine point eight ten divided by nine point eight gives me one point zero two I'll just round to two digits right over there so that gives me one point zero two seconds so our change in time so this right over here is one point zero two so our change in time delta T well I'm using lowercase now but I can make this all lowercase is equal to one point zero two one point zero two seconds now how do we use this information to figure out how far this thing travels well if we assume that it maintains its horizontal component of its velocity the whole time we just assume we can just multiply that times our change in time and we'll get the total displacement in the horizontal direction so to do that we need to figure out this horizontal component which we didn't do yet so this is the component of our velocity in the X direction or the horizontal direction once again we break out a little bit of trigonometry we this this side is adjacent to the angle so the adjacent over hypotenuse is the cosine of the angle cosine of an angle is adjacent over hypotenuse so we get cosine cosine of 30 degrees I just want to make sure I color code it right cosine of 30 degrees is equal to the adjacent side is equal to the adjacent side which is the magnitude of our horizontal component is equal to the adjacent side over the hypotenuse over ten meters per second multiply both sides by 10 meters per second you get the magnitude of our adjacent side the color transitioning is difficult the magnitude of our adjacent side is equal to 10 meters per second is equal to 10 meters per second times the cosine times the cosine of 30 degrees and you might not remember the cosine of 30 degrees you could use a calculator for this or you can just if you do remember it you know that it's the square root of 3 over 2 square root of 3 over 2 so to figure out the actual component I'll still have to get a calculator out if I want and that well I don't have to use it do it just yet because I have 10 times the square root of 3 over 2 which is going to be 10 divided by 2 is 5 so it's going to be five times the square root of three meters per second so if I want to figure out its entice the entire horizontal displacement so let's think about it this way the horizontal displacement that's what we're trying to figure out the horizontal displacement s for displacement is going to be equal to the average velocity in the in the X direction or in the horizontal direction and that's just going to be this five square roots of three meters per second because it doesn't change so it's going to be five I'm going to do that same color it's going to be the five square roots of three meters per second times the change in time times how long it is in the air and we figure that out it's one point zero two seconds times one point zero two seconds the seconds cancel out with seconds and we'll get an answer in meters and now we just get our calculator out to figure it out so we have five times the square root of three times one point zero two gives us eight point eight three meters just around it so this is going to be equal to this is going to be equal to this is going to be oh sorry this is going to be equal to eight point eight is that with the number I got eight point eight three eight point eight three meters and we're done in the next video I'm going to try to show you another way of solving for this delta T to show you really that there's multiple ways to solve this it's a little bit more complicated but it's also a little bit more powerful if we don't start at an end at the same elevation