Physics 4A Chapter 4: Motion in Two and Three Dimensions There is nothing either good or bad, but thinking makes it so. William Shakespeare It s not what happens to you that determines how far you will go in life; it is how you handle what happens to you. Zig Ziglar Reading: pages 58 76 Outline: motion in two and three dimensions position and displacement average and instantaneous velocity average and instantaneous acceleration projectile motion equations of projectile motion example problems uniform circular motion relative motion (read on your own) Problem Solving The problems of this chapter deal mainly with the definitions of average and instantaneous velocity and acceleration, with projectile motion, and with uniform circular motion. To calculate the average velocity you need to know the position at the beginning and end of a time interval. To calculate the velocity you need to know the position as a function of time. To calculate the average acceleration you need to know the velocity at the beginning and end of a time interval. To calculate the acceleration you need to know the velocity as a function of time. When you read a projectile motion problem, you should be able to identify two events, just as you did for one-dimensional problems. Take the time to be 0 for one of them, the launching of the projectile, for example. Take the y axis to be vertically upward and the x axis to be horizontal in the plane of the motion. The coordinates and velocity components are x 0, y 0, v 0x, and v 0y for the event at time 0. Let t be the time of the other event. The coordinates and velocity components are x, y, v x, and v y for that event. Identify the known and unknown quantities, then solve x = x 0 + v 0x t, y = y 0 + v 0y t ½ gt 2, and v y = v 0y -gt simultaneously for the unknowns. As an alternative you might use v y 2 - v 0y 2 = -2g(y-y 0 ) instead of the equation for y or the equation for v y. Remember v x = v 0x. All centripetal acceleration problems are solved using a = v 2 /r. This equation contains three quantities. Two must be given, either directly or indirectly. Remember that the acceleration vector points toward the center of the circle if the speed is constant. Sometimes the period T of
the motion is given. Remember that v = 2πr/T, where r is the radius of the circular orbit. You can use this expression to eliminate v in favor of r or r in favor of v in the expression for the centripetal acceleration. All relative motion problems are essentially vector addition problems. Identify the two reference frames of interest. Identify the velocities in the equation v PA = v PB + v BA, write the equation in component form and solve for the unknown quantities. Sometimes the equation must be rewritten in terms of magnitudes and angles rather than in terms of components. Questions and Example Problems from Chapter 4 Question 1 The figure below shows three situations in which identical projectiles are launched from the ground (at the same level) at identical initial speeds and angles. The projectiles do not land on the same terrain, however. Rank the situations according to the final speeds of the projectiles just before they land, greatest first. Question 2 An airplane flying horizontally at a constant speed of 350 km/h over level ground releases a bundle of food supplies. Ignore the effect of the air on the bundle. What are the bundle's initial (a) vertical and (b) horizontal components of velocity? (c) What is its horizontal component of velocity just before hitting the ground? (d) If the airplane's speed were, instead, 450 km/h, would the time of fall be larger, smaller, or the same?
Problem 1 The velocity v 2 of a particle moving in the xy plane is given by v = (6.0t 4.0 t ) iˆ+ 8.0 ĵ, with v in meters per second and t (> 0) in seconds. (a) What is the acceleration when t = 3.0 s? (b) When (if ever) is the acceleration zero? (c) When (if ever) is the velocity zero? (d) When (if ever) does the speed equal 10 m/s? Problem 2 An electron s position is given by ˆ 2 ˆ ˆ r = (3.00 t) i (4.00 t ) j+ 2.00k, with t in seconds and r in meters. (a) In unit-vector notation, what is the electron s velocity vt ()? At t = 2.00 s, what is v (b) in unit-vector notation and as (c) a magnitude and (d) an angle relative to the +x-axis.
Problem 3 A rifle is aimed horizontally at a target 30 m away. The bullet hits the target 1.9 cm below the aiming point. What are (a) the bullet's time of flight and (b) its speed as it emerges from the rifle? Problem 4 A small ball rolls horizontally off the edge of a tabletop that is 1.20 m high. It strikes the floor at a point 1.52 m horizontally from the table edge. (a) How long is the ball in the air? (b) What is its speed at the instant it leaves the table?
Problem 5 The punter on a football team tries to kick a football so that it stays in the air for a long hang time. (a) If the ball is kicked with an initial velocity of 25.0 m/s at an angle of 60.0 o above the ground, what is the hang time? (b) How far does the ball travel before it hits the ground? Problem 6 A rocket is fired at a speed of 75.0 m/s from ground level, at an angle of 60.0 o above the horizontal. The rocket is fired toward an 11.0 m high wall, which is located 27.0 m away. By how much does the rocket clear the top of the wall?
Problem 7 A golf ball is struck at ground level. The speed of the golf ball as a function of the time is shown in the figure below, where t = 0 at the instant the ball is struck. (a) How far does the golf ball travel horizontally before returning to ground level? (b) What is the maximum height above ground level attained by the ball? Problem 8 A plane, diving with a constant speed at an angle of 53.0 with the vertical, releases a projectile at an altitude of 730 m. The projectile hits the ground 5.00 s after being released. (a) What is the speed of the aircraft? (b) How far did the projectile travel horizontally during its flight? What were the (c) horizontal and (d) vertical components of its velocity just before striking the ground?
Problem 9 In the figure below, a baseball is hit at a height h = 1.00 m and then caught at the same height. It travels alongside a wall, moving up past the top of the wall 1.00 s after it is hit and then down past the top of the wall 4.00 s later, at distance D = 50.0 m farther along the wall. (a) What horizontal distance is traveled by the ball from hit to catch? What are the (b) magnitude and (c) angle (relative to the horizontal) of the ball s velocity just after being hit? (d) How high is the wall?
Problem 10 A sprinter running on a circular track has a velocity of constant magnitude 9.2 m/s and a centripetal acceleration of magnitude 3.8 m/s 2. What are (a) the track radius and (b) the period of circular motion? Problem 11 (a) What is the magnitude of the centripetal acceleration of an object on Earth's equator owing to the rotation of Earth? (b) What would the period of rotation of Earth have to be for objects on the equator to have a centripetal acceleration with a magnitude of 9.8 m/s 2?