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1.7 Falling objects  (Page 7/10)

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Problems&Exercises

Assume air resistance is negligible unless otherwise stated.

Calculate the displacement and velocity at times of (a) 0.500, (b) 1.00, (c) 1.50, and (d) 2.00 s for a ball thrown straight up with an initial velocity of 15.0 m/s. Take the point of release to be y 0 = 0 size 12{y rSub { size 8{0} } =0} {} .

(a) y 1 = 6 . 28 m size 12{y rSub { size 8{1} } =6 "." "28 m"} {} ; v 1 = 10 . 1 m/s size 12{v rSub { size 8{1} } ="10" "." "1 m/s"} {}

(b) y 2 = 10 . 1 m size 12{y rSub { size 8{2} } ="10" "." "1 m"} {} ; v 2 = 5 . 20 m/s size 12{v rSub { size 8{2} } =5 "." "20 m/s"} {}

(c) y 3 = 11 . 5 m ; v 3 = 0 .300 m/s size 12{v rSub { size 8{3} } =0 "." "300"" m/s"} {}

(d) y 4 = 10 .4 m ; v 4 = − 4 .60 m/s size 12{v rSub { size 8{4} } = - 4 "." "60"" m/s"} {}

Calculate the displacement and velocity at times of (a) 0.500, (b) 1.00, (c) 1.50, (d) 2.00, and (e) 2.50 s for a rock thrown straight down with an initial velocity of 14.0 m/s from the Verrazano Narrows Bridge in New York City. The roadway of this bridge is 70.0 m above the water.

A basketball referee tosses the ball straight up for the starting tip-off. At what velocity must a basketball player leave the ground to rise 1.25 m above the floor in an attempt to get the ball?

v 0 = 4 . 95 m/s size 12{v rSub { size 8{0} } =4 "." "95 m/s"} {}

A rescue helicopter is hovering over a person whose boat has sunk. One of the rescuers throws a life preserver straight down to the victim with an initial velocity of 1.40 m/s and observes that it takes 1.8 s to reach the water. (a) List the knowns in this problem. (b) How high above the water was the preserver released? Note that the downdraft of the helicopter reduces the effects of air resistance on the falling life preserver, so that an acceleration equal to that of gravity is reasonable.

A dolphin in an aquatic show jumps straight up out of the water at a velocity of 13.0 m/s. (a) List the knowns in this problem. (b) How high does his body rise above the water? To solve this part, first note that the final velocity is now a known and identify its value. Then identify the unknown, and discuss how you chose the appropriate equation to solve for it. After choosing the equation, show your steps in solving for the unknown, checking units, and discuss whether the answer is reasonable. (c) How long is the dolphin in the air? Neglect any effects due to his size or orientation.

(a) a = − 9 . 80 m/s 2 size 12{a= - 9 "." "80 m/s" rSup { size 8{2} } } {} ; v 0 = 13 . 0 m/s size 12{v rSub { size 8{0} } ="13" "." "0 m/s"} {} ; y 0 = 0 m size 12{y rSub { size 8{0} } ="0 m"} {}

(b) v = 0 m/s . Unknown is distance y to top of trajectory, where velocity is zero. Use equation v 2 = v 0 2 + 2 a y − y 0 size 12{v rSup { size 8{2} } =v rSub { size 8{0} } rSup { size 8{2} } +2a left (y - y rSub { size 8{0} } right )} {} because it contains all known values except for y , so we can solve for y size 12{y} {} . Solving for y size 12{y} {} gives

v 2 − v 0 2 = 2 a y − y 0 v 2 − v 0 2 2 a = y − y 0 y = y 0 + v 2 − v 0 2 2 a = 0 m + 0 m/s 2 − 13.0 m/s 2 2 − 9.80 m /s 2 = 8.62 m alignl { stack { size 12{v rSup { size 8{2} } - v rSub { size 8{0} } rSup { size 8{2} } =2a left (y - y rSub { size 8{0} } right )} {} #{ {v rSup { size 8{2} } - v rSub { size 8{0} } rSup { size 8{2} } } over {2a} } =y - y rSub { size 8{0} } {} # y=y rSub { size 8{0} } + { {v rSup { size 8{2} } - v rSub { size 8{0} } rSup { size 8{2} } } over {2a} } =0`m+ { { left (0`"m/s" right ) rSup { size 8{2} } - left ("13" "." 0`"m/s" right ) rSup { size 8{2} } } over {2 left ( - 9 "." "80"`"m/s" rSup { size 8{2} } right )} } =8 "." "62"`m {}} } {}

Dolphins measure about 2 meters long and can jump several times their length out of the water, so this is a reasonable result.

(c) 2 . 65 s size 12{2 "." "65 s"} {}

A swimmer bounces straight up from a diving board and falls feet first into a pool. She starts with a velocity of 4.00 m/s, and her takeoff point is 1.80 m above the pool. (a) How long are her feet in the air? (b) What is her highest point above the board? (c) What is her velocity when her feet hit the water?

(a) Calculate the height of a cliff if it takes 2.35 s for a rock to hit the ground when it is thrown straight up from the cliff with an initial velocity of 8.00 m/s. (b) How long would it take to reach the ground if it is thrown straight down with the same speed?

Path of a rock being thrown off of cliff. The rock moves up from the cliff top, reaches a transition point, and then falls down to the ground.

(a) 8.26 m

(b) 0.717 s
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Source:  OpenStax, Sample chapters: openstax college physics for ap® courses. OpenStax CNX. Oct 23, 2015 Download for free at http://legacy.cnx.org/content/col11896/1.9
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