h1_6a - Chapter 1 Mathematical Prelude cbPhysicsIa18.doc...

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Homework Assignment 1 Solutions Problem 1) Understanding Components and Vectors A) You would more likely give your friends a number of blocks to go east and then a number of blocks to go north. What would these two numbers be? As can be seen in the diagram below, you would tell your friend to travel 3 blocks east and 2 blocks north. B) Consider the vector b with magnitude 4.00 m at an angle 23.5 o north of east. What is the x component b x of this vector? First, we need to convert our degrees into radians (so it will work in the on-line answer key). In general, to convert from degrees into radians, use the following conversion: rad = deg ⋅ 180 o rad = 23.5 o ⋅ 180 o = 0.410 rad Then, to get our answer, we simply use the trigonometric equation mention in part A: b x = b cos  rad b x = 4.00 m cos 0.410 rad b x = 3.66 m C) Consider the vector b with length 4.00 m m at an angle 23.5 o north of east. What is the y component b y of this vector? Similar to part B, our y component is found using trigonometry:
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b y = 4.00 m sin 0.410 rad b y = 1.59 m D) What is the length of the shadow cast on the vertical screen by your 10.0 cm hand if it is held at an angle of = 30 o above horizontal? This question is similar to part C. It is asking for the y component of a vector 10.0 cm long at an angle of = 30 o above the horizontal. The setup is similar: rad = 30 o ⋅ 180 o = 0.52 rad lengthof shadow = 10.0 cm sin 0.52 rad lengthof shadow = 5.0 cm E) What is the magnitude of a vector with components (15 m, 8 m)? To find the magnitude of a vector with the given components, assuming they form a right triangle with the vector, you simply need to use the Pythagorean Theorem: A 2 B 2 = C 2 . A and B are the legs and C is the hypotenuse. In this problem, we are given the components A and B and need to solve for C: v ∣= 15 m 2  8 m 2 ∣ v ∣= 17 m F) what is the angle above the x axis (I.e. “north of east”) for a vector with components (15 m, 8 m)? To find the angle, if given to components, we need to use the tangent trigonometric function: tan = opposite adjacent = arctan opposite adjacent = arctan 8 m 15 m = 28.1 o or = 0.49 rad
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Problem 2) Vector Addition Suppose that you are swimming in a river while a friend watches from the shore. In calm water, you swim at a speed of 1.25 m/s . The river has a current that runs at a speed of 1.00 m/s. Note that speed is the magnitude of the velocity vector. The velocity vector tells you both how fast something is moving and in which direction it is moving. Part A) If you are swimming upstream (i.e., against the current), at what speed does your friend on the shore see you moving? Since your vector is the opposite direction as the current, you simply subtract the magnitudes of the
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h1_6a - Chapter 1 Mathematical Prelude cbPhysicsIa18.doc...

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