v a = -9.8 m/s Projectile Motion Good practice problems in book: 3.3, 3.5, 3.7, 3.9, 3.31, 3.33, 3.43, 3.47, 3.51, 3.53, 3.55 Practice test Reminder: test Feb 8, 7-10pm! Email me if you have conflicts! Treat this practice test like the exam. 3 hours, closed book, non-graphing calculator, pencil and paper. This is your next homework (not graded). Do it ASAP - it will indicate how well you ll probably do on the exam. Your intuitive understanding of the Physical world Percent grasp of general Physics principles Note: all of you got an A. Tested your perception of basic Physics principles. We re going to work on fixing your misconceptions! For instance: everything falls at the same rate! Please ignore your ecampus grade for now!
Today! Figure 1. Dysfunctional projectile Decomposing X and Y movements. How to hit a target. How to go about Projectile Motion problems. Ignoring air resistance, what would be the path of motion if someone ran off of a cliff? Q Straight down almost immediately A B C Straight down after some time Straight out for a little while then gradually down. D Never completely straight down Three velocity vectors. v vx vy Three velocity vectors. v vx vy v = vx0 Starts with horizontal velocity vx0 And vertical velocity vy0 = 0 m/s.
Free body! v = vx0 ay = -9.8 m/s v = vx0 Free body! Q3 ay = -9.8 m/s Throughout this fall, what is the jumper s horizontal acceleration vector, ax? A. vx0/t B. 9.8 m/s C. 0 m/s D. Not given, so unknown Free body! v = vx0 vx = v x0 + axt ay = -9.8 m/s Acceleration needed to change velocity In free bodies, the only acceleration is in the y direction vx is always equal to vx0 (unless it hits something) v vx vy Free body! vx = vx0 t = 1s t = s vx = vx0 vx = vx0 t = 3s vx = vx0
v vx vy vy0 = 0 m/s +y t = 1s How far downward has the jumper fallen after 3 seconds? t = s Δx = v 0 t + ½ at v = v 0 + aδx t = 3s v vx vy vy0 = 0 m/s t = 1s +y t = s Δx = v 0 t + ½ at v = v 0 + aδx vy = vy0 -gt Δy = vy0t -½ gt t = 3s Generic Projectile Motion y motion: Same as vertical-only problem! x motion: Covers Δx distance at vx0 speed x and y: Motions are Independent Separation of vectors into components allows separations of equations into components: v = v x xo + a t Δx = v t + xo x 1 axt v = v y yo + a t Δy = v t + yo y 1 ayt ax = 0 m/s ay = ±9.8 m/s
Monkey Hunter The instant the hunter fires, the monkey will reflexively let go of the tree and drop to the ground. Where should the hunter aim? A. Above the monkey. B. At the monkey. C. Below the monkey. Q4 Monkey Hunter If there were no gravity, the bullet would hit the monkey because the monkey would not move. Monkey Hunter With gravity, the bullet and the monkey fall at the same rate: the rate of the acceleration of gravity, -9.8 m/s. The answer is the same no matter where the hunter is standing. Break up what you know in terms of the horizontal and vertical Prevents mistakes! X Y
After it leaves your hand, before it hits the ground or t vo = initial velocity vector θo = initial direction of velocity vector v y = 0 at top of trajectory v x = v xo remains the same throughout trajectory because there is no acceleration along the x-direction Strategy for Projectile Motion Problems Beyond this, projectile motion problems just take a lot of planning and thinking. Take your time and think about the set-up of the problem. What do I know? What s the first step? What s the next step? Strategy for Projectile Motion Problems The time will be the same for x and y parts of the question. If you don t have enough information for x or y components, solve for time. Δy Δx Throwing something off of a cliff (5 examples with increasing difficulty) 1. How much time does it take to fall?. How far from the base of the cliff does it hit the ground? (Need the time first) 3. How fast it is moving vertically when it hits the ground? (y component of final velocity) 4. What is the magnitude of its velocity when it hits the ground? 5. What is the angle that it hits the ground from the horizontal?
How much time to fall? (Think Vertical) We re talking about something falling, and that is vertical motion, so we will only use vertical ideas and numbers. Δx = v 0 t + ½ at v = v 0 + aδx How much time to fall? (Think Vertical) We re talking about something falling, and that is vertical motion, so we will only use vertical ideas and numbers. Δy = v oy t + 1/ at Δy = (16 sin 60) t +1/ (-9.8) t Quadratic eq. (at +bt+c=0): a=-4.9, b=13.85, c=- Δy =15 m t=[-b ± (b -4(a)(c))^ ½] /(a) =-13.85±(191.8-4(-4.9)(15))^ ½ ]/(-9.8) t= 1.41 ±.5= -0.84 s or 3.66 s How far from cliff base? (Think Horizontal) We know it was in the air for 3.66s (from the previous question), and it s moving at a constant speed in the x-direction the whole time (a x = 0). Δx = v 0 t + ½ at v = v 0 + aδx How far from cliff base? (Think Horizontal) We know it was in the air for 3.66s (from the previous question), and it s moving at a constant speed in the x-direction the whole time (a x = 0). Δx = v ox t+ ½ a x t where ax=0 Δx = v x t = (16 cos60º)(3.66s) Δx = 9m
Vertical speed when it lands? (Think Vertical) Asking for y component of final velocity. It s been accelerating down the whole time. We know that gravity is causing this acceleration, so we can figure out how fast it is going (vertically) when it hits the ground. Δx = v 0 t + ½ at v = v 0 + aδx Vertical speed when it lands? (Think Vertical) Asking for y component of final velocity. It s been accelerating down the whole time. We know that gravity is causing this acceleration, so we can figure out how fast it is going (vertically) when it hits the ground. v fy = v iy + a t v fy = 16 sin60º -9.8 x3.66 v fy = - m/s Definitely negative is good, since moving opposite from positive y direction Magnitude and Angle that it hits the ground? (Finally combine!) Final angle: which way is the ball going? How do we get is its final velocity vector? Δx = v 0 t + ½ at v = v 0 + aδx Magnitude and Angle that it hits the ground? (Finally combine!) Final angle: which way is the ball going? How do we get is its final velocity vector? v x v y v Vector sum!
Magnitude and Angle that it hits the ground? (Finally combine!) c = a + b = (8.0m/s) + (-m/s) c = 3.4 m/s tanθ = opp/adj = (- m/s) / (8m/s) Θ = tan -1 (-/8) = -70 The object is moving at 3m/s at an angle of 70 below the horizontal when it hits the ground. -