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Projectile (2D) Motion AP Physics B September 21, 2010 Trigonometry Review • Application of Trigonometry to Vectors Trigonometry R y q x y sin q R x cos q R y tan q x y = R sin q x = R cos q R2 = x2 + y2 Example 1: Find the height of a building if it casts a shadow 90 m long and the indicated angle is 30o. The height h is opposite 300 and the known adjacent side is 90 m. opp h tan 30 adj 90 m 0 h 300 h = (90 m) tan 30o 90 m h = 51.96 m Finding 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 q R q x y y = R sin q Finding components: Polar to Rectangular Conversions 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? N N R q x 400 m y 30o E y=? x=? The x-component (E) is ADJ: x = R cos q The y-component (N) is OPP: y = R sin q E 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? N Note: x is the side 400 m 30o y=? x=? E x = (400 m) cos 30o = +346 m, E adjacent to angle 300 ADJ = HYP x Cos 300 x = R cos q The x-component is: Rx = +346 m 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? N Note: y is the side 400 m 30o y=? x=? E opposite to angle 300 OPP = HYP x Sin 300 y = R sin q y = (400 m) sin 30o The y-component is: = + 200 m, N Ry = +200 m 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? N 400 m 30o Rx = Ry = +200 m E The x- and ycomponents are each + in the first quadrant +346 m Solution: The person is displaced 346 m east and 200 m north of the original position. Components of Motion • You 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. Components of Motion • If 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 Θ vy = v sin Θ Ex. Shooting Pool Shooting a game of pool, you hit a ball that causes it to move diagonally with a constant velocity of 0.50 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 ycomponents of motion. Projectile Motion • A 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 xand y-components and treat them separately. Horizontal 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) (a) How long is the ball in flight before striking the ground? (b) How far from the building does the ball strike the ground? Teeing Off A 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) (a) What is the maximum height reached by the ball? (b) What is its range? Hit 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? Homework: • Read Conceptual Physics (Hewitt) Chapters 2 & 3 • Do Problems 19—28 • From College Physics (Wilson) Do Problems…9, 15, 73, 74, 79, 83, 84, 85, 89, 91 Projectile Practice 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? 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? 3. 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? 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? 5. 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? 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? Homework: • Read Conceptual Physics (Hewitt) Chapters 2 & 3 • Do Problems 19—28 • From College Physics (Wilson) Do Problems…9, 15, 73, 74, 79, 83, 84, 85, 89, 91 Monkey and the Hunter A 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?