Assignment 1: Kinematics in 2 and 3 DimensionsDue: 2:00am on Saturday, September 4, 2010 Note: To understand how points are awarded, read your instructor's Grading Policy. [Switch to Standard Assignment View] Arrow Hits Apple An arrow is shot at an angle of above the horizontal. The arrow hits a tree a horizontal distance for the away, at the same height above the ground as it was shot. Use magnitude of the acceleration due to gravity. Part A Find , the time that the arrow spends in the air. Hint A.1 Find the initial upward component of velocity in terms of D. Hint not displayed Hint A.2 Find the time of flight in terms of the initial vertical component of velocity. Hint not displayed Hint A.3 Put the algebra together to find symbolically. Hint not displayed Answer numerically in seconds, to two significant figures. ANSWER: = 6.7 Correct Suppose someone drops an apple from a vertical distance of 6.0 meters, directly above the point where the arrow hits the tree. Part B How long after the arrow was shot should the apple be dropped, in order for the arrow to pierce the apple as the arrow hits the tree? Hint B.1 When should the apple be dropped Hint not displayed Hint B.2 Find the time it takes for the apple to fall 6.0 meters Hint not displayed Express your answer numerically in seconds, to two significant figures. ANSWER: = 5.6 Correct = Correct Circular Launch A ball is launched up a semicircular chute in such a way that at the top of the chute, just before it goes into free fall, the ball has a centripetal acceleration of magnitude 2 . Part A How far from the bottom of the chute does the ball land? Hint A.1 Speed of ball upon leaving chute Hint not displayed Hint A.2 Time of free fall Hint not displayed Hint A.3 Finding the horizontal distance Hint not displayed Your answer for the distance the ball travels from the end of the chute should contain ANSWER: = . Correct Problem 3.50 Spiraling Up. It is common to see birds of prey rising upward on thermals. The paths they take may be spiral-like. You can model the spiral motion as uniform circular motion combined with a constant upward velocity. Assume a bird completes a circle of radius 8.00 m every 5.00 s and rises vertically at a rate of 3.00 m/s. Part A Part A Find the speed of the bird relative to the ground. ANSWER: 10.5 m/s Correct Part B Find the magnitude of the bird's acceleration. ANSWER: 12.6 Correct Part C Find the direction of the bird's acceleration. ANSWER: 0.0 Correct above the horizontal Part D Find the angle between the bird's velocity vector and the horizontal. ANSWER: 16.6 Correct Problem 3.69 Two tanks are engaged in a training exercise on level ground. The first tank fires a paint-filled training round with a muzzle speed of 239 at an angle 10.8 above the horizontal while advancing toward the second tank with a speed of 16.5 speed of 34.0 relative to the ground. The second tank is retreating at a relative to the ground, but is hit by the shell. You can ignore air resistance and assume the shell hits at the same height above ground from which it was fired. Part A Find the distance between the tanks when the round was first fired. Take free fall acceleration to be ANSWER: = 9.80 . 1990 m Correct Part B Find the distance between the tanks at the time of impact. Take free fall acceleration to be = 9.80 . Take free fall acceleration to be ANSWER: = 9.80 . 2150 m Correct Battleship Shells A battleship simultaneously fires two shells toward two identical enemy ships. One shell hits ship A, which is close by, and the other hits ship B, which is farther away. The two shells are fired at the same speed. Assume that air resistance is negligible and that the magnitude of the acceleration due to gravity is . Part A What shape is the trajectory (graph of y vs. x ) of the shells? ANSWER: straight line parabola hyperbola The shape cannot be determined. Correct Part B For two shells fired at the same speed which statement about the horizontal distance traveled is correct? Hint B.1 Two things to consider Hint not displayed ANSWER: The shell fired at a larger angle with respect to the horizontal lands farther away. The shell fired at an angle closest to 45 degrees lands farther away. The shell fired at a smaller angle with respect to the horizontal lands farther away. The lighter shell lands farther away. Correct Consider the situation in which both shells are fired at an angle greater than 45 degrees with respect to the horizontal. Remember that enemy ship A is closer than enemy ship B. Part C Which shell is fired at the larger angle? Hint C.1 Consider the limiting case Hint C.1 Consider the limiting case Hint not displayed ANSWER: A B Both shells are fired at the same angle. Correct Part D Which shell is launched with a greater vertical velocity, ANSWER: ? A B Both shells are launched with the same vertical velocity. Correct Part E Which shell is launched with a greater horizontal velocity, ANSWER: ? A B Both shells are launched with the same horizontal velocity. Correct Part F Which shell reaches the greater maximum height? Hint F.1 What determines maximum height? Hint not displayed ANSWER: A B Both shells reach the same maximum height. Correct Part G Which shell has the longest travel time (time elapsed between being fired and hitting the enemy ship)? Hint G.1 Consider the limiting case Hint not displayed ANSWER: A B Both shells have the same travel time. Correct A Wild Ride A car in a roller coaster moves along a track that consists of a sequence of ups and downs. Let the x axis be parallel to the ground and the positive y axis point upward. In the time interval from to s, the trajectory of the car along a certain section of the track is given by , where Part A At Hint A.1 is the roller coaster car ascending or descending? is a positive dimensionless constant. How to approach the problem Hint not displayed Hint A.2 Find the vertical component of the velocity of the car Hint not displayed ANSWER: ascending descending Correct Part B Derive a general expression for the speed Hint B.1 of the car. How to approach the problem Hint not displayed Hint B.2 Magnitude of a vector Hint not displayed Hint B.3 Find the components of the velocity of the car Hint B.3 Find the components of the velocity of the car Hint not displayed Express your answer in meters per second in terms of ANSWER: = and . Correct Part C The roller coaster is designed according to safety regulations that prohibit the speed of the car from exceeding . Find the maximum value of allowed by these regulations. Hint C.1 How to approach the problem To comply with the regulations, the speed of the car cannot exceed the given safety limit at any time. Thus, you need to determine what the maximum value of the speed is and impose the condition that such a value cannot be greater than the safety limit. Hint C.2 Find the maximum value of the speed of the car in terms of . Given the expression found in Part B, find the maximum speed Hint C.2.1 Using the calculus Hint not displayed Hint C.2.2 Find the first derivative of the speed Hint not displayed Hint C.2.3 Find the time at which the speed reaches its maximum value Hint not displayed Express your answer in meters per second. ANSWER: = Answer not displayed Express your answer using two significant figures. ANSWER: = 1.7 Correct Shooting over a Hill A projectile is fired with speed at an angle from the horizontal as shown in the figure . Part A Find the highest point in the trajectory, Hint A.1 . Velocity at the top Hint not displayed Hint A.2 Which equation to use Hint not displayed Express the highest point in terms of the magnitude of the acceleration due to gravity , the initial velocity ANSWER: = , and the angle . Correct Part B What is the range of the projectile, Hint B.1 ? Find the total time spent in air Hint not displayed Hint B.2 Find Hint not displayed Express the range in terms of ANSWER: = , , and . Correct Consider your advice to an artillery officer who has the following problem. From his current postition, he Consider your advice to an artillery officer who has the following problem. From his current postition, he must shoot over a hill of height at a target on the other side, which has the same elevation as his gun. He knows from his accurate map both the bearing and the distance to the target and also that the hill is halfway to the target. To shoot as accurately as possible, he wants the projectile to just barely pass above the hill. Part C Find the angle Hint C.1 above the horizontal at which the projectile should be fired. How to approach the problem Hint not displayed Hint C.2 Set up the ratio Hint not displayed Express your answer in terms of ANSWER: = and . Correct Recall the following trigonometry formulas: , , and . In this case, since can draw a right triangle with , you as one of the angles, an "opposite" side of length , and an "adjacent" side of length . You can then use this triangle to find and , after you find the length of the hypotenuse using the Pythagorean Theorem. Part D What is the initial speed? Hint D.1 How to approach this part Hint not displayed Hint D.2 Find Hint not displayed Hint D.3 Find Hint not displayed Express ANSWER: in terms of , , and . = Correct Part E Find , the flight time of the projectile. Hint E.1 How to proceed Hint not displayed Express the flight time in terms of ANSWER: = and . Correct Uniform Circular Motion Learning Goal: To find the velocity and acceleration vectors for uniform circular motion and to recognize that this acceleration is the centripetal acceleration. Suppose that a particle's position is given by the following expression: . Part A Choose the answer that best completes the following sentence: The particle's motion at can be described by ____________. ANSWER: an ellipse starting at time an ellipse starting at time on the positive x axis on the positive y axis a circle starting at time a circle starting at time Correct The quantity on the positive x axis on the positive y axis is defined to be the angular velocity of the particle. Note that must have units of radians per second. If is constant, the particle is said to undergo uniform circular motion. Part B When does the particle first cross the negative x axis? Express your answer in terms of some or all of the variables ANSWER: = , , and . Correct Now, consider the velocity and speed of the particle. Part C Find the particle's velocity as a function of time. Hint C.1 Derivative of Hint not displayed Express your answer using unit vectors (e.g., , , and ANSWER: + , where and are functions of , ). = Correct Part D Find the speed of the particle at time . Hint D.1 Definition of the magnitude of a vector Hint not displayed Hint D.2 Complete an mportant trig identity Hint not displayed Express your answer in terms of some or all of the variables ANSWER: , , and . ANSWER: Correct Note that the speed of the particle is constant: . Part E Now find the acceleration of the particle. Express your answer using unit vectors (e.g., , , and ANSWER: = + , where and are functions of , ). Correct Part F Your calculation is actually a derivation of the centripetal acceleration. To see this, express the acceleration of the particle in terms of its position . Express your answer in terms of some or all of the variables ANSWER: = and . Correct Part G Now find the magnitude of the acceleration as a function of time. Express your answer in terms of some or all of the variables ANSWER: = , , and . Correct Part H Finally, express the magnitude of the particle's acceleration in terms of you obtained for the speed of the particle. and using the expression Express your answer in terms of one or both of the variables ANSWER: = and . Correct There are three important things to remember about centripetal acceleration: There are three important things to remember about centripetal acceleration: The centripetal acceleration is simply the acceleration of a particle going around in a circle. It has magnitude of either or . It is directed radially inward. Curved Motion Diagram The motion diagram shown in the figure represents a pendulum released from rest at an angle of 45 from the vertical. The dots in the motion diagram represent the positions of the pendulum bob at eleven moments separated by equal time intervals. The green arrows represent the average velocity between adjacent dots. Also given is a "compass rose" in which directions are labeled with the letters of the alphabet. Part A What is the direction of the acceleration of the object at moment 5? Hint A.1 How to approach the problem Hint not displayed Hint A.2 Definition of acceleration Hint not displayed Hint A.3 Change of velocity: a graphical interpretation Hint not displayed Enter the letter of the arrow with this direction from the compass rose in the figure. Type Z if the acceleration vector has zero length. ANSWER: A Correct Part B What is the direction of the acceleration of the object at moments 0 and 10? Hint B.1 Find the direction of the velocity What is the direction of the velocity of this object at moments 1 and 9? Enter the letters of the corresponding directions from the compass rose, separated by Enter the letters of the corresponding directions from the compass rose, separated by commas. Type Z if the velocity vector has zero length. ANSWER: directions at time step 1, time step 9 = D,B Correct Hint B.2 Definition of acceleration Acceleration is defined as the change in velocity per unit time. Mathematically, . Since velocity is a vector, acceleration is a vector that points in the direction of the change in the velocity. Hint B.3 Applying the definition of acceleration To find the acceleration at moment 0, subtract the (vector) velocity at moment 0 from the velocity at moment 1. Similarly, to find the acceleration at moment 10, subtract the (vector) velocity at moment 9 from the velocity at moment 10. Enter the letters corresponding to the arrows with these directions from the compass rose in the figure, separated by commas. Type Z if the acceleration vector has zero length. ANSWER: directions at time step 0, time step 10 = D,F Correct A Canoe on a River A canoe has a velocity of 0.300 flowing at 0.600 southeast relative to the earth. The canoe is on a river that is east relative to the earth. Part A Find the magnitude of the velocity Hint A.1 of the canoe relative to the river. How to approach the problem In this problem there are two reference frames: the earth and the river. An observer standing on the edge of the river sees the canoe moving at 0.300 , whereas an observer drifting with the river edge of the river sees the canoe moving at 0.300 current perceives the canoe as moving with velocity relative to the earth is known, you can determine magnitude of . , whereas an observer drifting with the river Since the velocity of the current in the river . Note that the problem asks for the Hint A.2 Find the relative velocity vector Hint not displayed Hint A.3 Find the components of the velocity of the canoe relative to the river Hint not displayed Express your answer in meters per second. ANSWER: = 0.442 Correct Part B Find the direction of the velocity of the canoe relative to the river. Hint B.1 How to approach the problem Hint not displayed Hint B.2 Find the direction of a vector given its components Hint not displayed Express your answer as an angle measured south of west. ANSWER: 28.7 degrees south of west Correct Problem 3.34 The Ferris wheel in the figure , which rotates counterclockwise, is just starting up. At a given instant, a passenger on the rim of the wheel and passing through the lowest point of his circular motion is moving at 3.00 m/s and is gaining speed at a rate of . Part A Find the magnitude of the passenger's acceleration at this instant. ANSWER: 0.814 Correct Part B Find the direction of the passenger's acceleration at this instant. ANSWER: 37.9 Correct to the right of vertical Problem 3.78 You are flying in a light airplane spotting traffic for a radio station. Your flight carries you due east above a highway. Landmarks below tell you that your speed is 59.0 relative to the ground and your air speed indicator also reads 59.0 . However, the nose of your airplane is pointed somewhat south of . east and the station's weather person tells you that the wind is blowing with speed 25.0 Part A In which direction is the wind blowing? Express your answer as an angle measured east of north. ANSWER: Answer not displayed east of north Score Summary: Your score on this assignment is 99.3%. You received 39.7 out of a possible total of 40 points.