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$\bullet$ Two archers shoot arrows in the same direction from the same place with the same initial speeds but at different angles. One shoots at $45^{\circ}$ above the horizontal, while the other shoots at $60.0^{\circ} .$ If the arrow launched at $45^{\circ}$ lands 225 $\mathrm{m}$ from the archer, how far apart are the two arrows when they land? (You can assume that the arrows start at essentially ground level.)

30 $\mathrm{m}$

Physics 101 Mechanics

Chapter 3

Motion in a Plane

Physics Basics

Motion Along a Straight Line

Motion in 2d or 3d

Newton's Laws of Motion

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03:28

Newton's Laws of Motion are three physical laws that, laid the foundation for classical mechanics. They describe the relationship between a body and the forces acting upon it, and its motion in response to those forces. These three laws have been expressed in several ways, over nearly three centuries, and can be summarised as follows: In his 1687 "Philosophiæ Naturalis Principia Mathematica" ("Mathematical Principles of Natural Philosophy"), Isaac Newton set out three laws of motion. The first law defines the force F, the second law defines the mass m, and the third law defines the acceleration a. The first law states that if the net force acting upon a body is zero, its velocity will not change; the second law states that the acceleration of a body is proportional to the net force acting upon it, and the third law states that for every action there is an equal and opposite reaction.

04:16

In mathematics, a proof is a sequence of statements given to explain how a conclusion is derived from premises known or assumed to be true. The proof attempts to demonstrate that the conclusion is a logical consequence of the premises, and is one of the most important goals of mathematics.

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so we know that one arrow is shot at it angle at 45 degrees and it goes 225 meters. That is the range s So we can safely say assume that delta y equals zero meters for both arrows, Which means that the only distance only distance from one arrow to the other arrow will be in the ex direction S o. We're simply trying to find Delta are so we first need to thought we first say that our equals the initial squared sine of tooth ada divided by gee, that's the That's the range formula. Ah, and we need to solve for the initial velocity because we don't know the initial velocity of this arrow, which would be the initial velocity of the other girl that a shot at 60 degrees. So we can say that 45 times to theta equals 45. So two times 45 degrees would be 90 degrees sign of 90 degrees would be one. So this is simply equalling our equals the initial squared over G. Because we can say that this term goes to one. At this point, we can say that the initial velocity is simply the square root of the G times are and this will be equal to acceleration due to gravity on Earth 9.8 and the range of 225 meters. And we find that the initial velocity of both arrows is going to be 46.96 meters per second. Now to find the range of the new ERO, we're going to say that data now equals 60 degrees and the initial equals 46.96 meters per second. We can say that our equals again the initial squared sine of tooth Ada divided by G and we can solve and say 46.96 squared time's sign of 120 degrees divided by 9.8 and we're getting that the new range is going to be 194.9 meters. So the question is asking us how far away from each other they are so delta are would be again 225 minus 194.9, and this is equaling 30.1 meters. So this would be the distance from the first arrow to the second arrow again the first arrow. Because it is shot at the maximum arrange. It has a higher range of 225 meters as opposed to the range of Thie Aeroshot. At 60 degrees on, Lee has a range of again 194.9 meters. Again, the difference would be 30.1 meters. That is the end of the solution. Thank you for watching.

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