MOTION OF A PROJECTILE Today’s Objectives: Students will be able to: Analyze the free-flight motion of a projectile Dynamics, Fourteenth Edition R.C Hibbeler In-Class Activities: • • • • • • • Check Homework Reading Quiz Applications Kinematic Equations for Projectile Motion Concept Quiz Group Problem Solving Attention Quiz Copyright ©2016 by Pearson Education, Inc All rights reserved READING QUIZ The downward acceleration of an object in free-flight motion is A) zero B) increasing with time C) 9.81 m/s D) 9.81 ft/s The horizontal component of velocity remains _ during a free-flight motion A) zero C) at 9.81 m/s B) constant Dynamics, Fourteenth Edition R.C Hibbeler D) at 32.2 ft/s Copyright ©2016 by Pearson Education, Inc All rights reserved APPLICATIONS A good kicker instinctively knows at what angle, θ, and initial velocity, vA, he must kick the ball to make a field goal For a given kick “strength”, at what angle should the ball be kicked to get the maximum distance? Dynamics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved APPLICATIONS (continued) A basketball is shot at a certain angle What parameters should the shooter consider in order for the basketball to pass through the basket? Distance, speed, the basket location, … anything else? Dynamics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved APPLICATIONS (continued) A firefighter needs to know the maximum height on the wall she can project water from the hose What parameters would you program into a wrist computer to find the angle, θ, that she should use to hold the hose? Dynamics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved MOTION OF A PROJECTILE (Section 12.6) Projectile motion can be treated as two rectilinear motions, one in the horizontal direction experiencing zero acceleration and the other in the vertical direction experiencing constant acceleration (i.e., from gravity) Dynamics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved MOTION OF A PROJECTILE (Section 12.6) For illustration, consider the two balls on the left The red ball falls from rest, whereas the yellow ball is given a horizontal velocity Each picture in this sequence is taken after the same time interval Notice both balls are subjected to the same downward acceleration since they remain at the same elevation at any instant Also, note that the horizontal distance between successive photos of the yellow ball is constant since the velocity in the horizontal direction is constant Dynamics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved KINEMATIC EQUATIONS: HORIZONTAL MOTION Since ax = 0, the velocity in the horizontal direction remains constant (v x = vox) and the position in the x direction can be determined by: x = xo + (vox) t Why is ax equal to zero (what assumption must be made if the movement is through the air)? Dynamics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved KINEMATIC EQUATIONS: VERTICAL MOTION Since the positive y-axis is directed upward, ay = – g Application of the constant acceleration equations yields: vy = voy – g t y = yo + (voy) t – ½ g t 2 vy = voy – g (y – yo) For any given problem, only two of these three equations can be used Why? Dynamics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved EXAMPLE I Given: vA and θ Find: Horizontal distance it travels and vC Plan: Apply the kinematic relations in x- and ydirections Solution: Using vAx = 10 cos 30 and vAy = 10 sin 30 We can write vx = 10 cos 30 vy = 10 sin 30 – (9.81) t x = (10 cos 30) t y = (10 sin 30) t – ½ (9.81) t Since y = at C = (10 sin 30) t – ½ (9.81) t Dynamics, Fourteenth Edition R.C Hibbeler ⇒ t = 0, 1.019 s Copyright ©2016 by Pearson Education, Inc All rights reserved EXAMPLE I (continued) Only the time of 1.019 s makes sense! Velocity components at C are; vCx = 10 cos 30 = 8.66 m/s → vCy = 10 sin 30 – (9.81) (1.019) = -5 m/s = m/s ↓   =10 m/s Horizontal distance the ball travels is; x = (10 cos 30) t x = (10 cos 30) 1.019 = 8.83 m Dynamics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved EXAMPLE II Given: Projectile is fired with vA=150 m/s at point A Find: The horizontal distance it travels (R) and the time in the air Plan: Dynamics, Fourteenth Edition R.C Hibbeler How will you proceed? Copyright ©2016 by Pearson Education, Inc All rights reserved EXAMPLE II Given: Projectile is fired with vA=150 m/s at point A Find: The horizontal distance it travels (R) and the time in the air Plan: Establish a fixed x, y coordinate system (in this solution, the origin of the coordinate system is placed at A) Apply the kinematic relations in x- and y-directions Dynamics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved EXAMPLE II (continued) Solution: 1) Place the coordinate system at point A Then, write the equation for horizontal motion + → xB = xA + vAx tAB where xB = R, xA = 0, vAx = 150 (4/5) m/s Range, R, will be R = 120 tAB 2) Now write a vertical motion equation Use the distance equation +↑ yB = yA + vAy tAB – 0.5 g tAB where yB = – 150, yA = 0, and vAy = 150(3/5) m/s We get the following equation: –150 = 90 tAB + 0.5 (– 9.81) tAB Solving for tAB first, tAB = 19.89 s Then, R = 120 tAB = 120 (19.89) = 2387 m Dynamics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved CONCEPT QUIZ In a projectile motion problem, what is the maximum number of unknowns that can be solved? A) B) C) D) The time of flight of a projectile, fired over level ground, with initial velocity V o at angle θ, is equal to? A) (vo sin θ)/g C) (vo cos θ)/g Dynamics, Fourteenth Edition R.C Hibbeler B) (2vo sin θ)/g D) (2vo cos θ)/g Copyright ©2016 by Pearson Education, Inc All rights reserved GROUP PROBLEM SOLVING I y Given: x A skier leaves the ski jump ramp at θA = 25 o and hits the slope at B Find: The skier’s initial speed vA Plan: Dynamics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved GROUP PROBLEM SOLVING I y Given: x A skier leaves the ski jump ramp at θA = 25 o and hits the slope at B Find: The skier’s initial speed vA Plan: Establish a fixed x,y coordinate system (in this solution, the origin of the coordinate system is placed at A) Apply the kinematic relations in x- and y-directions Dynamics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved GROUP PROBLEM SOLVING I (continued) Solution: Motion in x-direction: Using xB = xA + vox(tAB) ⇒ (4/5)100 = + vA (cos 25°) tAB tAB= 80 vA (cos 25°) = 88.27 vA Motion in y-direction: Using yB = yA + voy(tAB) – ½ g(tAB) – 64 = + vA(sin 25°) { 88.27 } vA – ½ (9.81) { 88.272 } vA vA = 19.42 m/s tAB= (88.27 / 19.42) = 4.54 s Dynamics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved GROUP PROBLEM SOLVING II Given: The golf ball is struck with a velocity of 80 ft/s as shown Find: Distance d to where it will land y Plan: x Dynamics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved GROUP PROBLEM SOLVING II Given: The golf ball is struck with a velocity of 80 ft/s as shown Find: Distance d to where it will land y x Plan: Establish a fixed x, y coordinate system (in this solution, the origin of the coordinate system is placed at A) Apply the kinematic relations in x- and y-directions Dynamics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved GROUP PROBLEM SOLVING II (continued) Solution: Motion in x-direction: Using xB = xA + vox(tAB) y ⇒ d cos10 = + 80 (cos 55) tAB x tAB = 0.02146 d Motion in y-direction: Using yB = yA + voy(tAB) – ½ g(tAB) ⇒ d sin10 = + 80(sin 55)(0.02146 d) – ½ 32.2 (0.02146 d)2 ⇒ = 1.233 d – 0.007415 d2 d = 0, 166 ft Only the non-zero answer is meaningful Dynamics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved ATTENTION QUIZ A projectile is given an initial velocity vo at an angle φ above the horizontal The velocity of the projectile when it hits the slope is the initial velocity vo A) less than C) greater than B) equal to D) None of the above A particle has an initial velocity vo at angle φ with respect to the horizontal The maximum height it can reach is when A) φ = 30° B) φ = 45° C) φ = 60° D) φ = 90° Dynamics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved End of the Lecture Let Learning Continue Dynamics, Fourteenth Edition R.C Hibbeler Copyright ©2016 by Pearson Education, Inc All rights reserved  ... experiencing constant acceleration (i.e., from gravity) Dynamics, Fourteenth Edition R. C Hibbeler Copyright ©20 16 by Pearson Education, Inc All rights reserved MOTION OF A PROJECTILE (Section 12. 6) ... OF A PROJECTILE (Section 12. 6) Projectile motion can be treated as two rectilinear motions, one in the horizontal direction experiencing zero acceleration and the other in the vertical direction... ball is constant since the velocity in the horizontal direction is constant Dynamics, Fourteenth Edition R. C Hibbeler Copyright ©20 16 by Pearson Education, Inc All rights reserved KINEMATIC EQUATIONS: