2Trigonometry Review Application of Trigonometry to Vectors y = R sin q xRqx = R cos qR2 = x2 + y2
3Example 1: Find the height of a building if it casts a shadow 90 m long and the indicated angle is 30o.90 m300The height h is opposite 300 and the known adjacent side is 90 m.hh = (90 m) tan 30oh = m
4Finding Components of Vectors A component is the effect of a vector along other directions. The x and y components of the vector (R,q) are illustrated below.x = R cos qy = R sin qxyRqFinding components:Polar to Rectangular Conversions
5Example 2: A person walks 400 m in a direction of 30o N of E Example 2: A person walks 400 m in a direction of 30o N of E. How far is the displacement east and how far north?x = ?y = ?400 m30oENxyRqNEThe x-component (E) is ADJ:x = R cos qThe y-component (N) is OPP:y = R sin q
6Note: x is the side adjacent to angle 300 Example 2 (Cont.): A 400-m walk in a direction of 30o N of E. How far is the displacement east and how far north?x = ?y = ?400 m30oENNote: x is the side adjacent to angle 300ADJ = HYP x Cos 300x = R cos qThe x-component is:Rx = +346 mx = (400 m) cos 30o= +346 m, E
7Note: y is the side opposite to angle 300 Example 2 (Cont.): A 400-m walk in a direction of 30o N of E. How far is the displacement east and how far north?x = ?y = ?400 m30oENNote: y is the side opposite to angle 300OPP = HYP x Sin 300y = R sin qThe y-component is:Ry = +200 my = (400 m) sin 30o= m, N
8The x- and y- components are each + in the first quadrant Example 2 (Cont.): A 400-m walk in a direction of 30o N of E. How far is the displacement east and how far north?Rx = +346 mRy = +200 m400 m30oENThe x- and y- components are each + in the first quadrantSolution: The person is displaced 346 m east and 200 m north of the original position.
9Components of MotionYou can use the same approach to describe motion— and the motion doesn’t have to be in straight lines.Using vectors will allow you to analyze the behavior of batted balls, planets circling the Sun, and even electrons in atoms.Think of ball moving in x- and y-directions simultaneously.That is, it has a velocity in the x-direction (vx) and a velocity in the y-direction (vy) at the same time.The combined velocity components describe the actual motion of the ball.
10Components of MotionIf a constant velocity (v) in a direction at an angle (Θ) relative to the x-axis is given,then the velocities in the x- and y- directions are obtained by resolving, or breaking down, the velocity vector into components of motion:vx = v cos Θvy = v sin Θ
11Ex. Shooting PoolShooting a game of pool, you hit a ball that causes it to move diagonally with a constant velocity of m/s at an angle of 37° relative to the x-axis. Find how far it travels in 3.0 s by using x- and y- components of motion.
12Projectile MotionA familiar example of two-dimensional, curvilinear motion is the motion of objects thrown or projected by some means (cannon, baseball bat, etc.)Projectile Motion is considered to be in free fall, so the only acceleration of a projectile is due to gravity.We can use vector components to analyze projectile motion. We simply break up the motion into its x- and y-components and treat them separately.
13Horizontal Projection A ball is projected from a height of 25.0 m above the ground and is thrown with an initial horizontal velocity of 8.25 m/s (fig 3.16)How long is the ball in flight before striking the ground?How far from the building does the ball strike the ground?
14Teeing OffA golf ball is hit off the tee with an initial velocity of 30.0 m/s at an angle of 35° to the horizontal (fig 3.17)What is the maximum height reached by the ball?What is its range?
15Hit or Miss?A young girl standing on a bridge throws a stone with an initial velocity of 12 m/s at a downward angle of 45° to the horizontal, in an attempt to hit a block of wood floating in the river below.If the stone is thrown from a height of 20 m and it reaches the river when the block is 13 m from the bridge, does the stone hit the block?
16Homework: Read Conceptual Physics (Hewitt) Chapters 2 & 3 Do Problems 19—28From College Physics (Wilson) Do Problems…9, 15, 73, 74, 79, 83, 84, 85, 89, 91
181. A ball is thrown straight up with a speed of 12. 5 m/s 1. A ball is thrown straight up with a speed of 12.5 m/s. (a) How high does it go and (b) how much time does it take to get there?
192. A volkswagon runs straight off a cliff traveling at a speed of 34 2. A volkswagon runs straight off a cliff traveling at a speed of 34.5 m/s. If the cliff is 12.5 m high, how far horizontally does the car travel before it smashes into the ground?
203. A stealth bomber on a training mission drops one of its bombs from a height of 3,500m. The bomb travels a horizontal distance of 1.25 km. What was the plane’s horizontal speed?
214. An arrow is launched with a velocity of 88. 7 m/s at an angle of 33 4. An arrow is launched with a velocity of 88.7 m/s at an angle of 33.0° to the horizontal. How far does the arrow travel?
225. A brick is thrown upward from the top of a building at an angle of 25° with an initial speed of 15 m/s. It strikes the ground below. If the brick is in flight for 3.0s, how tall is the building?
236. A ball is thrown at an angle of 43° to the horizontal 6. A ball is thrown at an angle of 43° to the horizontal. It travels a distance of 75m in 2.3s. (a) What was its original velocity? (b) How high did it go?
24Homework: Read Conceptual Physics (Hewitt) Chapters 2 & 3 Do Problems 19—28From College Physics (Wilson) Do Problems…9, 15, 73, 74, 79, 83, 84, 85, 89, 91
26Monkey and the HunterA zookeeper finds an escaped monkey hanging from a 4-meter high tree eating a banana. Aiming her tranquilizer gun at the monkey, the zookeeper kneels and aims 21.8° from the ground, 10 meters from the tree. The monkey thinks he is smart; as soon as the zookeeper fires her gun, the monkey releases from the tree. If the tranquilizer dart travels at 50.0m/s, will the zookeeper hit the monkey?