2012 P.A.P.A-2 FREE-RESPONSE QUESTIONS (Calc OK)

Transcription

1 1 P.A.P.A- FREE-RESPONSE QUESTIONS (Calc OK) 1) At the Jackson County Fair held each October, there are many games and attractions which launch things into the air. One game is a football toss where the object is to get the football to land on a nickel placed on the ground. Darla is the only contestant whose throw landed on the nickel perfectly. With her winning throw, she threw the ball from a height of 1.7 and with an upward velocity of 4 per and a forward velocity of 33 per. a) Write the parametric equations which models Darla s winning throw. b) What was the speed and the release angle of Darla s winning throw? c) How far was the Nickel placed from where Darla threw the football? Describe d) The fair offers a new car to anybody who hits the nickel (which Darla did) and then on their next throw is able to hit a dime which is 5 away. If Darla s nd throw had an upward velocity of 3 per, What angle and overall initial speed did the ball need to be thrown with in order to land on the dime and win the car?

2 1 P.A.P.A- FREE-RESPONSE QUESTIONS 1) At the Jackson County Fair held each October, there are many games and attractions which launch things into the air. One game is a football toss where the object is to get the football to land on a nickel placed on the ground. Darla is the only contestant whose throw landed on the nickel perfectly. With her winning throw, she threw the ball from a height of 1.7 and with an upward velocity of 4 per and a forward velocity of 33 per. a) Write the parametric equations which models Darla s winning throw. b) What was the speed and the release angle of Darla s winning throw? c) How far was the Nickel placed from where Darla threw the football? Describe d) The fair offers a new car to anybody who hits the nickel (which Darla did) and then on their next throw is able to hit a dime which is 5 away. If Darla s nd throw had an upward velocity of 3 per, What angle and overall initial speed did the ball need to be thrown with in order to land on the dime and win the car? 1: Units provided throughout a) Equation for the height: h t t t Equation for the forward direction: ( ) f ( t) 33t b) The football was thrown with a speed of 4.84 per and with a release angle of c) The football was in the air for s. During this time the football moved forward before falling to the ground and landing on the nickel. Therefore the nickel was horizontally away from where the football was released. d) The new parametric function to model the height of the football in this situation would be h( t) 4.9t 3t 1.7 and the ball would be in the air for s. For the ball to move horizontally 5 during this time, the forward velocity was per. Therefore, the football was thrown with a speed of 31.7 per and with a release angle of 74.93(8) o. 1: both vertical and forward motion equations are provided. 1: Speed : 1: release angle 1: considers t = sec : 1: states distance of : considers t= : forward velocity of m/sec 3: 1: determines release angle and speed

3 13 P.A.P.A- FREE-RESPONSE QUESTIONS (Calculator OK) 3. One of the attractions at the Wilson County Fair held every September is the human catapult. During this event, held nightly at 8pm, a trained acrobat is shot from a cannon which is set 3 off the ground into a deep pond which sits 7 away and is 14 long and is deep enough throughout the length of the pond for a safe landing (the acrobat is also a skilled diver and swimmer). Picture not drawn to scale Cannon Pond X 7m 84m a) For the initial testing a gigantic pumpkin was used (similar in weight to the acrobat to model the release speed accurately). If the pumpkin was in the air for 4.36 s and landed 65 away, what are the parametric functions to model the parabolic flight of this pumpkin? (Use h(t) for the vertical parametric and f(t) for the forward motion parametric). b) Another practice pumpkin was launched at an angle of 35 and a release speed of 6 per. Did the pumpkin land in the pond? Justify c) If the acrobat was launched with a horizontal velocity of per. What was the release angle necessary for the acrobat to land in the exact middle of the pond? A B C

4 13 P.A.P.A- FREE-RESPONSE QUESTIONS (Calculator OK) 3. One of the attractions at the Wilson County Fair held every September is the human catapult. During this event, held nightly at 8pm, a trained acrobat is shot from a cannon which is set 3 off the ground into a deep pond which sits 7 away and is 14 long and is deep enough throughout the length of the pond for a safe landing (the acrobat is also a skilled diver and swimmer). The initial set up had the cannon placed 3 high. Picture not drawn to scale Cannon Pond X 7m 84m a) For the initial testing a gigantic pumpkin was used (similar in weight to the acrobat to model the release speed accurately). If the pumpkin was in the air for 4.36 s and landed 65 away, what are the parametric functions to model the parabolic flight of this pumpkin? (Use h(t) for the vertical parametric and f(t) for the forward motion parametric). b) Another practice pumpkin was launched at an angle of 35 and a release speed of 6 per. Did the pumpkin land in the pond? Justify c) If the acrobat was launched with a horizontal velocity of per. What was the release angle necessary for the acrobat to land in the exact middle of the pond? a) h v (4.36) 4.9(4.36) v(4.36) Equation for the height: f ( t) 65 f (4.36) f h t t t ( ) Equation for the forward direction: f ( t) (6) t 1: Units provided throughout 1: both vertical and forward motion equations are provided. 1: Speed : 1: release angle vertical velocity sin(35) b) 6 vertical velocity & horizontal velocity cos(35) 6 horizontal velocity : considers t = sec : 1: states distance of Using h( t) 4.9t t 3, the ball was in the air for s The ball would have traveled, f( t) (3.3849) OR The pumpkin would have just missed the front edge of the pond as the pond was 7 away. c) The acrobat needed to be in the air for 3.85 s ( f ( t) 65 t ) If the acrobat was in the air for 3.85 s, this would require an initial vertical velocity of h(3.85) 4.9(3.85) v (3.85) 3) ( 1 vertical velocity release angle= tan tan horizontal velocity The release angle was thus 4.1(6) o 1: considers t= : forward velocity of m/sec 3: 1: determines release angle and speed

5 14 P.A.P.A- FREE-RESPONSE QUESTIONS P.A.P.A- FRQ Part A Time 3 minutes Number of problems A graphing calculator is required for these problems. ) The height of a thrown ball in a recent game is modeled by the function h, where h( t) 4.93t 7.8t 1.8 is measured in and t is measured in s and t. Assume the ground is at a height of for all parts of this question. a) Compute the value of Compute the value for when h() and state the meaning of this point in the context of the problem. ht ( ) b) Write the domain for function h. Give a reason for your answer and state the meaning of this point in the context of the problem. c) If the Distance the ball travels horizontally from the thrower is modeled by d( t) 9.5t,where measured in. How far horizontally has the ball traveled when the ball is at its maximum height? d) The function h is the equation which models the height of this ball while standing on earth. On Mars, the height p( t) 1.856t v t 1.5. If the ball on earth and the ball on mars of a ball is modeled with the function were to be thrown simultaneously. What would the initial upward velocity ( would hit the ground of its planet at the same time? v dt () is ) have to be so that each ball

6 14 P.A.P.A- FREE-RESPONSE QUESTIONS P.A.P.A- FRQ Part A Time 3 minutes Number of problems A graphing calculator is required for these problems. 3) The height of a thrown ball in a recent game is modeled by the function h, where h( t) 4.93t 7.8t 1.8 is measured in and t is measured in s and ground is at a height of for all parts of this question. a) Compute the value of h() and state the meaning of this point in the context of the problem. Compute the value for when ht ( ) t. Assume the and state the meaning of this point in the context of the problem. b) Write the domain for function h. Give a reason for your answer c) If the Distance the ball travels horizontally from the thrower is modeled by d( t) 9.5t, where measured in. How far horizontally has the ball traveled when the ball is at its maximum height? d) The function h is the equation which models the height of this ball while standing on earth. On Mars, the height p( t) 1.856t v t 1.5. If the ball on earth and the ball on mars of a ball is modeled with the function were to be thrown simultaneously. What would the initial upward velocity ( would hit the ground of its planet at the same time? a) h() h() is the height of the ball at time or the moment the ball was released from the throwers hand. h t t t ( ) t t.65 or 5.69sec t=5.69 represents the time when the ball would hit or land on the ground. b) t 5.69 before t=, the ball has not been thrown and after t=5.69 the ball is resting or bouncing on the ground in either case, the ball s height is not being modeled by h(t) outside of this domain. b 7.8 c) The time when the ball is at its maximum is a 4.93 d) d(.78197) 9.5(.78197) 81.37(81.373) v v v v p(5.69) v 1.18( or 1.181) dt () v ) have to be so that each ball 1: value of h() with meaning 3: 1: value of t when h(t)= 1: Justification / reasons is 1: domain statement : 1: Justification / reasons 1: considered t = : 1: Horizontal distance traveled 1: uses t =5.69 : 1: Solves for initial velocity with units