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At $t=0$ a batter hits a bascball with an initial spced of 28 $\mathrm{m} / \mathrm{s}$at a $55^{\circ}$ angle to the horizontal. An outficlder is 85 $\mathrm{m}$ fromthe batter at $t=0$ and, as seen from home plate, the line ofsight to the outficlder makes a horizontal angle of $22^{\circ}$ with the plane in which the ball moves (sce Fig. $64 ) .$ What speed and direction must the ficlder take to catch the ball at the sameheight from which it was struck? Give the angle with respect to the outficlder's line of sight to home plate.

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$6.8 \mathrm{m} / \mathrm{s}$, $61^{\circ}$

Physics 101 Mechanics

Chapter 3

Kinematics in Two or Three Dimensions; Vectors

Motion Along a Straight Line

Motion in 2d or 3d

Newton's Laws of Motion

Rotation of Rigid Bodies

Dynamics of Rotational Motion

Equilibrium and Elasticity

Simon Fraser University

University of Sheffield

University of Winnipeg

McMaster University

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In physics, a rigid body is an object that is not deformed by the stress of external forces. The term "rigid body" is used in the context of classical mechanics, where it refers to a body that has no degrees of freedom and is completely described by its position and the forces applied to it. A rigid body is a special case of a solid body, and is one type of spatial body. The term "rigid body" is also used in the context of continuum mechanics, where it refers to a solid body that is deformed by external forces, but does not change in volume. In continuum mechanics, a rigid body is a continuous body that has no internal degrees of freedom. The term "rigid body" is also used in the context of quantum mechanics, where it refers to a body that cannot be squeezed into a smaller volume without changing its shape.

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In physics, rotational dynamics is the study of the kinematics and kinetics of rotational motion, the motion of rigid bodies, and the about axes of the body. It can be divided into the study of torque and the study of angular velocity.

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okay. Our question says that at time T equals zero. A batter hits a baseball with an initial speed of 28 meters per second at an initial angle of 55 degrees to the horizontal. An outfielder is 85 meters from the batter. At T equals zero and has seen from home plate The line of the site of the outfielder makes an angle Horrors to the horizontal 22 degrees with the plane in which the ball moves. See figure 64. What speed and direction must the fielder take to catch the ball at the same height from which it was struck? Give the angle with respect to the outfielders line of sight of home plate. Okay, so I wrote down here a couple of things that we have. Well, the initial speed was 28 meters per second. The initial angle, which we're going to call they they're not was 55 degrees. Okay, we're also told that the outfielder write this that the Alfa Al fielders 50 85 meters from home plate. So I did note that over here, the 85 meters home plate I'm drawn a little red dot here This is home for the outfielder and then are here is the range the outfielder would have to cover. Okay. And ah, x Here in this X, here is the the era that shows the direction in which the outfielder must run. Okay, So since the ball is being caught at the same height from which he was struck, we can use the range formula from example three Dutch 10 to find the horizontal distance. The ball travels, which is are there that I did notice are Let's go ahead and do that. Okay, So the range formula is R Consul. Use black again. R is equal to the initial velocity V, not squared times assigned of two data, okay, divided by gravity. Okay. And maybe I should use different rotation. Will call this state of product data prime. Is that 22 degrees there? Okay, gravity is 9.8 meters per second on earth. Okay, so we played in all those values. We find that this equals 75 point 175 meters for the range. Oh, okay. And that's then a scene from above the location of home plate and the point where the ball must be caught in the initial location of the outfielder are shown in the diagram. There, right the dark air there. I'm sorry, the arrow that I indicated X, as I previously stated, is thie the length of the distance which the outfielder must run and you confined. This using the law of co signs is applied to a triangle. So we're going to find excusing the love co science where X is equal to the square root of In general. It's a squared plus B squared plus two a b co sign of the angle that's in general. But for our situation, let's plug in those values. It's are is a is R squared plus 85 meters squared plus two times are times 85 times Co sign here. Let's, uh, put some princes around 85 so you can tell that that's a multiplication. Good notation there. Time is 85 times that co sign of the angle of the England question. Here is the 22 degrees. Okay, and if you plug in for are the 75.175 that we found before you find that this X is equal to 32 went zero for eight meters. Okay, so now we found the distance that the fielder Al filler must come cover. And the angle fada at which the outfielder should run is found fromthe law of co sides. So now we're going to use the law of co signs which says that he sign of the angles. Data divided by, uh, ex is equal to, uh I'm sorry that this is the sine of the angle fatal prime Times X is equal to sine of the angle fada divided by our. So there's the loss of co signs. So now we want to solve for theta because that's what we want to find. We know everything else. Data is equal to the inverse sign of, uh, are divided by x times. Sign the fate of Prime. Okay, so if you play gin for our 75.175 meters, you plug in for X 32.0 for eight and you multiply that ratio by the sign of 22 degrees, you will find you'll find that that comes out to equal 61.49 degrees. Okay. And, uh, since what we are actually trying to find, uh, is, um, the will average velocity of the outfielder. We can assume that the outfielders time for running is the same as the time of the flight of the ball. The time the flight of the ball is found from the horizontal motion of the ball at a constant velocity. So we know the ball is going to travel. The, uh, start a new page. We know the ball is going to travel the distance of the range, which is going to be equal to the loss in the extraction times the time consult for time. That's what we want. Time is equal to the range divided by the velocity and the extraction of lost being The extraction is Veena times the co sign of state or not. Okay, so if we plug in what we have on the previous page for the range 75.175 for V, not 28 meters per second and fourth oedema famous, not 55 degrees, we find that the time is equal to 4.618 seconds. No, now that we have the time, we confined the average velocity which is going to be equal to the distance X defined by the time. So this is the average velocity of the runner and that is going to be for extra plug in 32.48 meters for time. You plug in what we just found. 4.6 20 seconds. You find that this is equal to 6.8 meters per second as the average velocity of the outfielder. Okay. And in the angle that he ran, which we found what was data is equal to 61.49 degrees relative to home plate speaking box said it is the angle we're asked to find. There's our solution.

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