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A placekicker is about to kick a field goal. The ball is 26.9 ${m}$ from the goalpost. The ball is kicked with an initial velocity of 19.8 ${m} / {s}$ at an angle $\theta$ above the ground. Between what two angles, $\theta_{1}$ and $\theta_{2},$ will the ball clear the $2.74-{m}$ -high crossbar? (Hint: The following trigonometric

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Physics 101 Mechanics

Chapter 3

Kinematics in Two Dimensions

Motion in 2d or 3d

University of Washington

Hope College

University of Winnipeg

Lectures

04:01

2D kinematics is the study of the movement of an object in two dimensions, usually in a Cartesian coordinate system. The study of the movement of an object in only one dimension is called 1D kinematics. The study of the movement of an object in three dimensions is called 3D kinematics.

10:12

A vector is a mathematical entity that has a magnitude (or length) and direction. The vector is represented by a line segment with a definite beginning, direction, and magnitude. Vectors are added by adding their respective components, and multiplied by a scalar (or a number) to scale the vector.

02:31

A placekicker is about to …

07:27

02:10

06:51

Football Kicker A football…

15:36

A football player kicks a …

04:48

A placekicker must kick a …

05:34

01:36

10:37

A place kicker must kick a…

So the question states that a fuel goal kicker can kick a ball at 19 0.8 meters per second and there is a field goal that is 26.29 26.9 meters away with the height of 2.74 meters. Uh, tall and we're trying to find at what two angles can the ball be kicked at so that it passes just threw the upright of the the field goal. So to do this, the first thing we should do is figure out a way to represent the time in terms of the theater angle. So we know that the horizontal component of this, um velocity vector is going to be so Visa Becks is going to be 19.8 times co signed data and we know this because coastline data is adjacent overhype on news where the adjacent is visa Bendixen the hypothesis 19.8 So we can rearrange it and get this equation. Now that we know V of X, we can use our Kinnah Matic equations which state that the velocity in the X direction multiplied by the time should be eagle toothy range of the projectile so we know that the velocity is 19.8 times co sign of data Times T should be equal to 26.9 meters, which means t the time that the project I was in the Air Force should be 26.9 divided by 19.8 times co sign of theater Now that we know this, we can use another Kinnah Matic equation too eso for the state of value that we want. So the one we're gonna use now is the one that states that the change in displacement in the Y direction is equal to the initial velocity in the Y direction times the time plus 1/2 times the acceleration times the time squared. So we know the change in displacement is going to be 2.74 meters. We know the initial velocity in the Y direction is 19.8 times signed the data. It's just the same thing. Similar calculation to the horizontal of lost e calculation. So in 19.8 time sign data Times T, which we calculated up here to be 26.9, divided by 19.8 times co sign of data plus 1/2 times the acceleration due to gravity, which is negative 9.8 meters per second squared times This time square, it's a 26.9 over in 19.8 co sign data and all this will be squared. So now that we have this, we can simplify it a little bit, so we're gonna have 2.74 is equal to. So this signing co sign will become tangent of Fada and these 19.8 will cancel out. So we'll just be left with 26 0.9 times the tangent of state. Uh, plus, um, 1/2 times negative, 9.8 times 26.9 squared over 19.8 squared times, one over co sine squared, which the same thing as seeking squared, which is also the same thing as one plus 10 tension square. So we'll have one plus tangent squared here of data. So now that we have this, we can simplify it a little bit further and get a, um, equation that's equal to zero. So when we I'm starting from the right and side here, so we're gonna have one plus 10 squared the, uh, times this number, which I simplified the one this 1/2 blah, blah, blah, blah, blah. It's going to be simplified into negative 9.0 for for to plus 26.9 times the attention of data minus 2.74 It's gonna be eagle Zero and we can distribute this one plus 10 square data. And when we do this, we will get the following equation. Get nine negative, 9.0 for for two can squared. They'd, uh, plus 26.9 10 data minus 11.7842 is equal to zero. And at this point, we can set our 10 of Fada equal to X, and we can solve it as a, uh, using quadratic formula to find the roots. Or we can just use a graphing calculator to find the roots. And so when we use a graphing calculator, we'll see that one of the routes is 10.534 and the other is 2.44 So tan of data will either eagle 0.534 or 2.44 So now if we want to find the angle fatal, we can just take the 10 inverse of both of these, so 10 in verse of 0.534 as well as the 10 inverse of 2.4 for And we'll get that thief state of values that work. Um, for this for the situation is 28 0.1 degrees as well as 67.71 degrees, and that's your final answer.

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