Such a life, with all vision limited to a Point and all motion to a Straight Line, seemed to me inexplicably dreary -Edwin Abbott Abbott (

Transcription

1 AP Physics Motion in 2-D Projectile and Circular Motion Introduction: In previous units, we studied motion along a straight line. Now we broaden our discussion to include motion along a curved path that lies in a plane. A few familiar examples are the flight of a thrown, a projectile shot from a gun and a ball whirled around in a circle at the end of a cord. To describe these motions, we need to combine the kinematics and vector addition lessons learned earlier. The vector quantities of velocity and acceleration will not always be parallel to each other. For an object to follow a curved path, a component of the acceleration vector must be perpendicular to the motion (velocity vector) of the object. This unit deals mostly with kinematics. Performance Objectives: Upon completion of the readings and activities in this unit and when asked to respond either orally or on a written test, you will: Describe the position of a particle by a position vector r and the displacement of the particle by the vector r. Describe its average velocity and instantaneous velocity by a velocity vector v. Find the horizontal and vertical components of the instantaneous velocity vector. Describe its average acceleration and instantaneous acceleration by an acceleration vector a. Resolve the instantaneous acceleration vector into components which are parallel and perpendicular to the motion. Analyze the motion of projectiles near the surface of the earth (that is, objects with a constant downward acceleration.) Display an understanding of the independence of the vertical and horizontal velocities of a projectile. Apply this knowledge in solving problems involving projectiles. Write the position functions for both the vertical and horizontal motions of a projectile. Illustrate an understanding of projectiles fired horizontally and at an angle by solving problems associated with such projectiles. Write the equation which describes the trajectory of the projectile where the vertical position is a function of the horizontal position and the initial velocity. Write an equation which describes the range of the projectile in terms of its initial velocity. List the equations for centripetal acceleration and centripetal force. Analyze the motion of objects in uniform and non-uniform circular motion. Describe how radial and tangential acceleration affect the motion of an object. Solve problems for circular motion in the horizontal and vertical planes. Describe projectile and circular motion in unit vector notation. Solve problems involving unit vectors. Textbook Reference: Tipler Physics, Chapter 3 Such a life, with all vision limited to a Point and all motion to a Straight Line, seemed to me inexplicably dreary -Edwin Abbott Abbott ( Projectile Motion: Projectiles in physics are objects that are launched into the air by an external net force and then allowed to travel through the air with only the forces of gravity and air resistance acting on it. The path that the projectile follows is called a trajectory. Remember that velocity is a vector quantity and vectors can be resolved into perpendicular components. For a projectile launched other than straight up or straight down, the object is moving in both the x and y directions at the same time. Recall that vectors add together to give a net result, but act independently of each other. Keep this in mind when analyzing the motion of a projectile. If air resistance is neglected, the horizontal component has no net force acting upon it and the vertical component has only the force of gravity acting upon it. The instantaneous velocity at any time can be found by adding the horizontal and vertical velocities using the Pythagorean Theorem. These are actually kinematic problems, so go back and find that old reliable list of kinematic formulas that we built up in the kinematics unit! Conceptual Questions: 1.) Suppose you are on a ledge in the dark and wish to estimate how high above the ground the ledge is. So you drop a stone over the edge and hear it strike the ground in one second. About how high is the ledge above the ground? Could you estimate the height of the ledge above the ground with the stone if you weren t close enough to the edge to be able to drop the stone straight down? If so, how and why? If not, why not?

2 2.) A sailor drops his pocketknife from the top of a mast on a ship sailing eastward at 7 m/s. The mast is 21 m high. Where does the knife hit the deck? 3.) A football is kicked into an oncoming wind. a.) Sketch its path (trajectory) of travel. b.) How would its trajectory look different if air resistance could be ignored? Make a second sketch of this new trajectory. c.) for this second sketch, draw vectors representing the x and y components of velocity at five different locations along the football s trajectory. I.) When it leaves the ground. II.) When it s halfway up. III.) When it s at the peak of its trajectory. IV.) When it s halfway down. and V.) At the instant before it hits the ground. 4.) A ball is thrown vertically upward with a velocity of 24 m/s from a railroad flatcar moving horizontally with a velocity of 4.0 m/s. Describe the path of the ball as seen a.) by an observer on the flatcar, and b.) by an observer on the ground nearby. 5.) Why does a high jumper approach the jump at a relatively slow running speed where as a long jumper runs as fast as possible for the jump? Exercises and Problems: 6.) A stone is thrown horizontally at a speed of 10.0 m/s from the top of a cliff 78.4 m high. a.) How long does it take the stone to reach the bottom of the cliff? b.) How far from the base of the cliff does the stone strike the ground? 4 s 40 m 7.) A plane drops a raft to the survivors of a shipwreck. The plane is flying at a height of 1960 m and at a speed of 90.0 m/s. The raft lands next to the survivors. How far away from the shipwreck was the plane when the raft was dropped m 8.) Divers at Acapulco dive from a cliff that is 61 m high. If the rocks below the cliff extend outward for 23 m, what is the minimum horizontal velocity a diver must have to clear them? 6.5 m/s 9.) A stone is thrown horizontally from the top of a cliff m high. If the stone lands 40.0 m from the base of the cliff, a.) How long did it take the stone to fall to the base? b.) With what horizontal velocity was the stone thrown? 5 s 8 m/s 10.) A cannon ball is fired horizontally with a velocity of 350 m/s from the top of a cliff 85 m high. a.) In what time will it strike the plane at the foot of the hill? b.) At what distance from the foot of the cliff will it strike? c.) With what velocity will if strike? 4.2 s 1470 m 352 m/s 11.) Some folks claim that bullets fired by some high powered rifles travel in a straight line path without dropping. This is nonsense, for all bullets fired from any rifle drop beneath a straight line path a vertical distance given by y = (gt 2) /2. Explain why for small velocities the curved trajectory of a bullet is apparent, while for higher velocities the curve is less apparent. 12.) An arrow is fired directly (horizontally) at the bull s eye of a target 60.0 m away. The arrow has a speed of 89 m/s when it leaves the bow. When it is fired, the arrow is 1.0 m above the ground. How far short (in front of) of the target does it stride the ground? 19.8 m 13.) A world s record for a long jump of 8.9 meters was set by Bob Beamon in high-altitude Mexico City in the 1968 Olympics. This record stood for over twenty years when it was broken by Mike Powell, who jumped two inches farther on August 30, (Carl Lewis also broke the record that day, same event, but Powell s distance was better.) Beamon was not always given full credit for his record because some people felt that he had an advantage in Mexico City. What two factors relating to the high altitude favor the long jump? 14.) Why does a hunter raise the barrel of his rifle when aiming at a distant target? If he aims directly at a target m away, by how much will he miss the target (how far below the intended mark) if the muzzle velocity of the bullet is m/s? m Projectiles at an angle 15.) A projectile is fired with a speed of 196 m/s at an angle of 60 degrees with the horizontal. Calculate a.) the vertical velocity and the horizontal velocity of the projectile. b.) the time the projectile is in the air, and c.) the horizontal distance the projectile travels. 170 m/s 98 m/s 35 s 3430 m 16.) A projectile is fired at an angle of 53 degrees with the horizontal. The speed of the projectile is 200 m/s. Calculate a.) the time the shell remains in the air, and b.) the horizontal distance it travels s 3912 m

3 17.) A famous motorcyclist plans to jump across a canyon, 0.35 km wide. To do this, the cyclist plans to leave a 30-degree ramp on one side of the canyon at a speed of 19.6 m/s. If the motorcycle can attain this speed, where does the cycle touch down? Assuming that the top of the ramp is at the same level as the opposite side of the canyon, does the motorcycle reach the other side of the canyon on the jump? If it does not reach the other side, then at what initial velocity would the motorcycle need to leave the same ramp to make the jump a success? 18.) a.) Show that two projectiles fired at complimentary angles will have the same range. b.) What angle will produce the maximum range? 19.) A baseball is hit at 30 m/s at an angle of 53 with the horizontal. Immediately, an outfielder runs 4 m/s toward the infield and catches the ball at the same height it was hit. What was the original distance between the batter and the fielder? 108 m 20.) A broad jumper takes off at an angle of 20 degrees above the horizontal and jumps 0.6 m high. a.) What is his forward velocity? b.) How far does he jump? 9.42 m/s 6.6 m 21.) While standing on an open bed of a truck moving at 35 m/s, an archer sees a duck flying directly overhead. The archer shoots an arrow straight up at the duck and misses. The arrow leaves the bow with a vertical velocity of 98 m/s. a.) How long does it remain in the air? b.) The truck maintains a constant speed of 35 m/s and does not change its direction. What horizontal distance does the arrow travel while it is in the air? c.) Where does the arrow finally land (ignore air resistance)? More Projectiles Fired at an Angle. In the problems above, the projectile began and ended its trajectory at the same height. Now the trajectory of the projectile will end above or below its original starting height. 22.) A cat jumps off of a table that is 1.0 m high. The initial velocity of the cat in 3.0 m/s at an angle of 37 degrees above the horizontal. How far out from the edge of the table does the cat strike the floor? 1.6 m 23.) A diver takes off with a speed of 8.0 m/s from a diving board 3.0 m high, at an angle of 30 above the horizontal. How much later does he strike the water? 1.3 s 24.) A pilot cuts loose two of his fuel tanks in an effort to gain altitude. At the time of release, he was 140 m above the ground and traveling upward at an angle of 30 degrees above the horizontal, with a speed of 86 m/s. How long were the tanks in the air? 11.3 s 25.) A batted baseball leaves the bat at an angle of 37 above the horizontal and with a velocity of 40.0 m/s. The ball is caught by a spectator in the outfield bleachers at a horizontal distance of 128 m from home plate. How far above the level at which the ball was struck was it caught? 17.7 m 26.) A projectile is fired into the air from the top of a m cliff above the valley. Its initial velocity is 67 m/s at an angle of 60 with the horizontal. Neglecting air resistance, where does the projectile land? m 27.) A ball thrown by a boy in the street is caught 2.00 seconds later by another boy on the porch of a house 14.7 m away and 4.9 m above the street level. What was the speed of the ball and the angle above the horizontal at which it was thrown? 14.3 m/s 28.) A large boulder rests on a cliff m above a small village in such a position that if it should roll off, it would leave with a speed of 50.0 m/s at an angle of 30 degrees below the horizontal. There is a pond, diameter m, with its edge m from the base of the cliff. a.) How long with the boulder be in the air? b.) What will the horizontal component of its velocity by when it hits? c.) A physics student says that the boulder will land in the pond. Is he right? What will the boulder s range be? 6.8 s v x = 43 m/s v y = 92 m/s 101 m/s 296 m Circular Motion: Circular motion results when a net force constantly acts at a right angle to the direction in which the mass is moving. This constant net force perpendicular to the motion of the body is called the centripetal force. Both the centripetal acceleration (a c = v 2 /r) and the centripetal force (F c = m a c = m v 2 /r) are always directed toward the center of the circle. If a body travels through the same circle more than once, then circular motion can be an example of periodic motion. Uniform Circular Motion occurs when the speed of the object in the circular path is constant. The time it takes the mass to complete one complete circle is known as the period (T). Frequency (f) on the other hand is the number of circular paths / cycles / events / vibrations / oscillations completed by the mass in a

4 specific amount of time (i.e. I watch NHL Hockey 7 times a week is an example of the frequency at which I watch NHL Hockey). If the specific amount of time at which these cycles have taken place in is measured in seconds, then the unit of frequency is Hertz (Hz) which translates to per second or 1/s. Period is the inverse of Frequency or: T = 1/f. 29.) What is the centripetal acceleration of a 2.0 kg object moving in a circular path of 20.0 m radius with a speed of 20.0 m/s? b.) What is the direction of the centripetal acceleration acting on the object? 20 m/s 2 30.) It takes a kg racing car 10.0 s to travel at a uniform speed around a circular racetrack of 50.0 m radius. a.) What is the acceleration of the car? b.) How can this car be accelerating if it is traveling at a uniform speed? 19.7 m/s 31.) A child twirls a yo-yo about his head in a horizontal circle with a radius of 0.80 m. a.) If the yo-yo make one complete revolution each second, what is the centripetal acceleration of the yo-yo? b.) If the child increases the speed of the yo-yo to 2.0 rev/s, what must be the centripetal acceleration on the yo- yo? c.) What is the ratio of the answer in (b) to (a)? d.) Why? 31.5 m/s m/s 2 4:1 32.) The radius of the moon s orbit about the earth is about 3.6 x 10 8 m. The moon s period is 2.3 x 10 6 seconds (27.3 days). Find the centripetal acceleration of the moon m/s 2 33.) What is the acceleration of the tip of the minute hand of a watch if the hand is 2.0 cm long? 6.1 x 10-8 m/s 2 34.) An athlete twirls a 78 kg hammer tied the end of a rope in a horizontal circle with a radius of 1.3 m. The hammer moves at the rate of 1.0 rev/s. What is the centripetal acceleration of the hammer? 51.3 m/s 2 35.) A carnival merry-go-round has a 10.0 m radius and rotates about a vertical axis at a constant speed. A passenger standing on the edge of the marry-goround experiences an acceleration of 1.3 m/s 2 toward the center of the merry-go-round. a.) What is the tangential speed of the passenger? B.) How long does it take the merry-go-round to make one revolution? 3.6 m/s 17.4 s Motion along a curved path where the speed and direction are changing. 36.) A particle moves along a circular path having a radius of 2.0 m. At an instant when the speed of the particle is equal to 3.0 m/s and changing at a rate of 5.0 m/s 2, what is the magnitude of the total acceleration of the particle? 6.7 m/s 2 37.) A car travels in a flat circle of radius R. At a certain instant, the velocity of the car is 20 m/s north, and the total acceleration of the car is 2.5 m/s 2 at 37 degrees south of west. What is the radius of the circle? Is the speed of the car increasing or decreasing? 200 m decreasing 38.) What is the instantaneous speed of a particle moving in a circle of radius 2.0 m at the instant that the particle s total acceleration is 6.0 m/s 2 and its tangential acceleration is 4.4 m/s 2? 2.9 m/s Motion in a Plane Unit Vector Notation. Try to visualize the motion of the particle form the information given. If you can do this, it will be easier to solve the problem? 39.) A particle starts from the origin at t = 0 s with a velocity of 8.0 j m/s and moves in the xy plane with a constant acceleration of (4.0 i j) m/s 2. At the instant that the x-coordinate of the particle is 29 m, what is the value of its y-coordinate? 45 m 40.) A ball is shot from the ground into the air. At a height of 9.1 m, its velocity is observed to be 7.6 i j in meters per second. a.) To what maximum height does the ball rise? b.) What total horizontal distance does the ball travel? c.) What are the magnitude and direction of the ball s velocity just before it hits the ground? 11 m 23 m 16.4 m/s 41.) A stunt pilot performs a circular dive of radius m moving in a counterclockwise direction. At the bottom of the dive, the pilot s acceleration is (20 i + 50 j) m/s 2. What is his instantaneous speed? 200 m/s

5 42.) The position of a particle as it moves in the xyplane is a function of time as given by r(t) = (6t) i + (-5t 2 ) j, where r is in meters and t is in seconds. a.) What is the position of the particle when t = 0 s and t = 3 s? b.) What is the displacement of the particle between t = 0 and t = 3 seconds? c.) What is the average velocity during the first three seconds? d.) What is the instantaneous velocity of the particle at t = 3 seconds? e.) What is the average acceleration during the first three seconds? f.) What is the instantaneous acceleration at t = 2 seconds? r(0) = 0 m 43.) A particle moves in the xy-plane in such a way that its position varies in time according to the expression r(t) = (4 + 2t 3t 2 ) i + (20t 5t 3 ) j where r is in meters and t is in seconds. a.) What is the instantaneous velocity of the particle at t = 1 s? b.) What is the instantaneous acceleration at t = 2 s? v(1) = (-4) i = (5) j a(2) = (-6) i + (-60) j 44.) A rabbit runs across a high school parking lot on which a set of coordinate axes has been drawn. The coordinates of the rabbit s position as functions of time (t) are given by: x = -0.3t t + 28 and y = 0.22t 2 9.1t + 30, with t in seconds and x and y in meters. a.) At t = 30 s, what is the rabbit s position vector (r) in unit vector notation and as a magnitude and an angle? b.) Graph the rabbit s path for t = 0 s to t = 50 s. (Use your graphing calculator in the parametric mode. x min = -300, x max = 75, y mini = -75, y max = 100) c.) What is r from t = 0 to t = 30 s? d.) Find the velocity (v) at time t = 30 s, in unit vector notation and as a magnitude and an angle. e.) Find the acceleration (a) at time t = 30 s, in unit vector notation and as a magnitude and an angle.