CHAPTER 6 PROJECTILE MOTION 1 Basic principle of analyzing projecting motion Independency of vertical and horizontal motion 2 A simple case: Horizontally projected motion An angry bird is fired horizontally at 6m/s from a cliff of height 10m. Calculate (a) the time of flight; (b) the horizontal distance travelled; (c) the speed of the angry bird when it hit the ground. Page 1
[HKALE] Three bombs are released froma bomber flying horizontallywith constant velocity to theright. They are released fromrest (relative to the bomber)one by one at one-secondintervals. Neglecting airresistance, which of thefollowing diagrams correctlyshows the positions of thebomber and the three bombs ata certain instant? [HKALE] A small object is thrown horizontally towards wall 1.2m away. It hits the wall 0.8 m below its initial horizontallevel. At what speed does the object hit the wall?(neglect air resistance.) Page 2
[HKALE (modified)] Inside a room, a small ball is projected horizontally witha speed of 1.2 m/s from the point P on a wall as shown. It hitsthe floor at Q and rebounds to the point R on the opposite wall.(neglecting airresistance and friction.) (a) Find the time of flight along the path PQR. (b) Find the height PS. (c) Find the speed of the ball at R, assuming the collision at Q is perfectly elastic. (d) At what angle (to the horizontal) does the ball hit the floor at Q? Page 3
3 General projectile motion [Example] 1 A long jump athlete takes off at a speed of 5ms and 30 o to the ground. Neglecting air resistance, find (a) the horizontal distance travelled when he reaches the ground; (b) the maximum height he can reach. Page 4
[Derivation] For a general projectile with initial speed u at angle θ, without air resistance, show that (a) the time of flight is 2u sinθ t =. g (b) the range is u 2 sin 2θ d = g (c) the equation of the trajectory is 2 gx y = + x tanθ. 2 2 2u cos θ Page 5
[Angry Bird] An angry bird, 10 m above the ground, is to be projected at angle 30 o to hit the target at horizontal distance 55 m from it and 12 m above the ground. Find the initial speed in order to hit the target. [Basketball] A basketball player is going to shoot from 3m in front of the target at 2.5 m from the ground. If the ball is shot from 2 m above the ground, and the initial speed of the ball is 8.5 m/s, what should the projection angle from the horizontal be? Page 6
[HKALE] [Table tennis] In a game of table tennis, the ball is struck when it is at C, which is 0.40 m vertically above the edge A ofthe table. Immediately after it is struck, the ball moves with a horizontal velocity v. It is then just passes thenet, hits the table at D and reaches the highest point E as shown below. The table is 2.70 m long and thenet is 0.15m high. Neglecting the effect of air resistance, calculate i) the value of v ii) the speed of the ball just before it hits the table. iii) If point E is at 0.25 m above the table, draw the graph of vertical velocity, vy, of the ball against time from C to E. Take downward as positive. (tc, tdand teon the time axis are the times when the ball is at C, D and E respectively. Page 7
4 The effect of air resistance on projectile trajectory [Derivation] Draw the expected trajectory when air resistance is NOT neglected Think before you draw: 1. When will the effect of air resistance become important? 2. What will be the change in maximum height and range? 3. Will the trajectory remain symmetric? FINAL REMARKS Serving as a good example of how physics theory can be applied in daily life, the study of projectile motion shows how one can combine basic theories of uniform motion, free fall, mathematical understanding of quadratic functions to give good model of daily life problem. In simple terms, all features of projectile motion can be understood by just a simple idea: the independency of vertical and horizontal motion. On problem solving, students are expected to perform calculations by treating vertical and horizontal motion in separate manner. Students are not encouraged to memorize the results of general projectile motion, but you should be able to derive by your own. Page 8