Displacement vs. Distance Suppose that an object starts at rest and that the object is subject to the acceleration function t

Transcription

1 MTH 54 Mr. Simond cla Diplacement v. Ditance Suppoe that an object tart at ret and that the object i ubject to the acceleration function t a() t = 4, t te over the time interval [,1 ]. Find both the diplacement and the ditance traveled by the object over that time interval. Aume that the length unit i cm and that the time unit i econd. Aume that the poition function i r. t t The velocity function i r () t = t, te e + 1. Two poible poition function are: 3 t t r() t = t 5, te e + t t t r() t = t 1, te e + t (hown in red in Figure 1 over t 1.5) (hown in blue in Figure 1 over t 1.5) Note that the ditance travelled and the diplacement vector are completely independent of the tarting point. Note that the curve i almot linear, o the ditance travelled and the magnitude of the diplacement vector will be nearly equal. Figure 1 Projectile Motion: Section 1.4 1

2 MTH 54 Mr. Simond cla Projectile Motion Introductory Problem Miy ha iue with her third grade teacher, Mr. Katz. One day Miy could no longer contain herelf and whipped out her truty pea hooter. After working up a good pit ball, Miy loaded her hooter, tuck it in her mouth at an angle of 65 relative to the floor, took a deep breath and let the ball fly. The end of the hooter wa three feet above the ground at the time the ball exited the hooter. The ball hit Mr. Katz in the back of the head five feet above the ground and 1 feet from the point at which the flight began. Find the initial peed of the ball. Figure 1: Student ketch of Miy Incident Background If we let g repreent the contant acceleration due to earth gravity, ignore inconvenient truth like air reitance, a non pherical globe, mountain and valley, etc., and define r() t to be the poition of the pitball t econd after it take flight, then the acceleration function i contantly r () t =, g (auming that we ue a coordinate ytem in uch a way that the ĵ component increae a the pitball move upward). r t = r t dt = g dt = gt + C. The velocity function, then, i () ( ),, If we aume that the initial velocity i v = v1, v ( ) ( ) ( ), then: r =, g + C = v C = v r t =, gt + v, v = v, gt + v Now, () () r t = r t dt = v1, g t + v dt = v1t, g t + vt + C1. If we aume that the initial poition of the pitball i =,, then: 1 r( ) = v ( ), g ( ) + v ( ) + C = C = Thi give u: 1 1 r t v t gt v t v t gt v t () = 1, + +, = 1, + + P rojectile Motion: Section 1.4

3 MTH 54 Mr. Simond cla If an object i in free flight ubject only to a planetary gravitational contant g, and the object wa launched from an initial poition =, at an initial velocity v = v1, v, then the poition of the object t time unit after launch i given by the function: Furthermore, if the initial peed i 1 r() t = v t, gt + v t + 1 v v in ( θ ) ft/ =. On earth, let ue g = 3 or g = 9.8 m/. v peed unit and the initial trajectory angle i θ, then v v ( θ ) = co and 1 Projectile Motion in a vacuum on a planet with gravitational contant g. g = = = r t = ( f ) ft/ O 3.1, 3 ft, θ 65, 1,5 Miy aumption v i the unknown initial peed of the pitball (ft/) t i the unknown total flight time of the ball () f Miy variable definition Projectile Motion: Section 1.4 3

4 MTH 54 Mr. Simond cla Baketball player Shaquille O'Neal attempted a hot while tanding ft from the baket. It "the Shaq" hot from a height of 7 ft and the ball reached a maximum height of 1 ft before wihing through the 1 ft high baket, what wa the initial peed of the ball? g = = r t = r t = ft/ 3.1, 7 ft, ˆj 1,1 Shaq aumption ( m) ft, ( f ) v i the unknown initial peed of the ball (ft/) θ i the unknown initial trajectory angle (degree) t i the unknown total flight time of the ball () f t i the unknown flight time until maximum height () m Analyi Since there are four unknown ( v,,, θ t m equation. Shaq variable definition and t f ), we will ultimately need to come up with four 4 P rojectile Motion: Section 1.4

5 MTH 54 Mr. Simond cla Projectile Motion Practice Problem 1. A ball roll off of a level tabletop 4 feet above the ground at the peed of 5 ft/ec. With what peed doe the ball hit the ground?. Nancy Lopez hit a golf ball from the tee with an initial peed of 15 ft/ec at an angle of 45 with repect to the horizontal. How far from the tee doe the ball land? 3. Sammy Soa hit a ball in the Atrodome three feet above the plate and at an angle of 4 with repect to the horizontal. The ball jut clear the 9 foot fence located 4 feet from home plate. Find the peed at which the ball began it flight and the peed at which the ball cleared the wall. 4. Fire break out on the roof of a foot building that i 5 ft by 5 ft at both the bae and top of the building. A fireperon i tanding at the midpoint relative to one ide of the bae aiming a hoe at the roof of the building. The top of the hoe i 4 feet above ground level and 15 feet from the bae of the building. Water i hooting out of the end of the hoe at a peed of 4 ft/ec. At what angle i the hoe being held if the water i landing mack dab in the middle of the roof? 5. Steve Young threw a pa at a 45 angle from a height of 6.5 ft with an initial peed of 5 ft/ec. At the nap, Jerry Rice began running directly downfield and caught the pa at the 6 yard line 6 feet above the ground. Mr. Young wa at the 5 yard line when he received the nap. How far back from the line of crimmage wa Mr. Young when he threw the ball? Figure : Football Field line of crimmage ball caught at 6 yard line Projectile Motion: Section 1.4 5

6 MTH 54 Mr. Simond cla Anwer to Practice Problem 1. The ball hit the ground with a peed of about ft/ec.. The ball hit the ground about feet from the tee. 3. The initial peed of the ball wa about ft/ec and the ball cleared the wall at about 13.6 ft/ec. 4. The hoe wa at an angle of about 66.7 with repect to the horizontal. 5. Mr. Young threw the ball about 6.6 feet behind the 5 yard line 6 P rojectile Motion: Section 1.4