Motion of Projectiles

More documents

Recommendations

Info

$When an integer, or a fraction of integers, is used in an equation, the ...$

70 Chapter 3 Motion in a Plane Conceptual Example 3.6 Monkey and Hunter An inexperienced hunter aims and shoots an arrow straight at a coconut that is being held by a monkey sitting in a tree (Fig. 3.22). At the same instant that the arrow leaves the bow, the monkey drops the coconut. Neglecting air resistance, does the arrow hit the coconut, the monkey, or neither? Strategy and Solution If there were no gravity, the arrow would fly straight to the monkey and coconut (along the dashed line from the bow to the monkey on the branch in Fig. 3.22). Since gravity gives the dropped coconut and the released arrow the same constant acceleration downward, they each fall the same vertical distance below the positions they would have had with no gravity. The coconut falls as shown by the dashed red line; the distance fallen at 0.25-s intervals is marked along a vertical axis. At the same time, the arrow drops below the blue dashed line by the amounts marked along its indicated trajectory at 0.25-s intervals. The arrow ends up hitting the coconut no matter what the initial velocity of the arrow. The higher the velocity of the arrow, the sooner they meet and the shorter the vertical distance that the coconut falls before being hit. Discussion An experienced hunter would have aimed above the initial position of the coconut to compensate for the amount his arrow would drop during the time of flight; he would have missed the dropping coconut but might have hit the monkey unless the monkey jumped down to retrieve the coconut. Conceptual Practice Problem 3.6 Changes in Position and Velocity for Consecutive Arrows An arrow is shot into the air. One second later, a second arrow is shot with the same initial velocity. While the two are both in the air, does the difference in their positions (r 2 – r 1 ) stay constant or does it change with time? Does the difference in their velocities (v 2 – v 1 ) stay constant or does it change with time? 0.3 m t = 0 s t = 0.25 s 1.2 m t = 0.50 s 2.8 m t = 0.75 s 1.2 m 2.8 m 0.3 m t = 0.25 s t = 0.50 s t = 0.75 s 4.9 m t = 1.00 s Figure 3.22 A monkey drops a coconut at the very instant an arrow is shot toward the coconut. In each quarter second, the coconut and arrow have fallen the same distance below where their positions would be if there were no gravity.
3.6 Velocity Is Relative; Reference Frames 71 Example 3.7 A Bullet Fired Horizontally A bullet is fired horizontally from the top of a cliff that is 20.0 m above a long lake. If the muzzle speed of the bullet is 500.0 m/s, how far from the bottom of the cliff does the bullet strike the surface of the lake? Ignore air resistance. Strategy We need to find the total time of flight so that we can find the horizontal displacement. The bullet is starting from the high point of the parabolic path because v iy = 0. As usual in projectile problems, we choose the y- axis to be the positive vertical direction. Known: ∆y = –20.0 m; v iy = 0; v ix = 500.0 m/s. To find: ∆x. Solution The vertical displacement through which the bullet falls is 20.0 m. The relationship between ∆y and ∆t is ∆y = 1 2 (v fy + v iy ) ∆t Substituting v iy = 0 and v fy = v iy + a y ∆t = a y ∆t yields ∆y = 1 2 a y(∆t) 2 ⇒∆t = 2 ∆y a y The horizontal displacement of the bullet is ∆x = v ix ∆t = v ix 2 ∆y a y = 500.0 m/s × 2 × (– 2 0 .0 m) – 9. 80 2 = 1.01 km m/s Discussion How did we know to start with the y- component equation when the question asks about the horizontal displacement? The question gives v ix and asks for ∆x. The missing information needed is the time during which the bullet is in the air; the time can be found from analysis of the vertical motion. We neglected air resistance in this problem, which is not very realistic. The actual distance would be less than 1.01 km. Practice Problem 3.7 Bullet Velocity Find the horizontal and vertical components of the bullet’s velocity just before it hits the surface of the lake. At what angle does it strike the surface? At the beginning of the chapter, we asked why the clam does not fall straight down when the gull lets go. The gull is flying horizontally with the clam, so the clam has the same horizontal velocity as the gull. When the gull lets go, the net force on the clam is downward due to gravity. The clam falls toward Earth, but since a x = 0 the clam retains the same horizontal component of velocity as the gull. Therefore, the clam is a projectile starting at the top of its parabolic trajectory. 3.6 VELOCITY IS RELATIVE; REFERENCE FRAMES The idea of relativity arose in physics centuries before Einstein’s theory. Nicole Oresme (1323–1382) wrote that motion of one object can only be perceived relative to some other object. Until now, we have tacitly assumed in most situations that displacements, velocities, and accelerations should be measured in a reference frame attached to Earth’s surface—that is, by choosing an origin fixed in position relative to Earth’s surface and a set of axes whose directions are fixed relative to Earth’s surface. After learning about relative velocities, we will take another look at this assumption. Relative Velocity Suppose Wanda is walking down the aisle of a train moving along the track at a constant velocity (Fig. 3.23). Imagine asking, “How fast is Wanda walking?” This question is not well defined. Do we mean her speed as measured by Tim, a passenger on the train, or her speed as measured by Greg, who is standing on the g – round and looking into the train as it passes by? The answer to the question “How fast?” depends on the observer. Figure 3.24 shows Wanda walking from one end of the car to the other during a time interval ∆t. The displacement of Wanda as measured by Tim—her displacement relative to the train—is ∆r WT = v WT ∆t. During the same time interval, the train’s displacement relative to the ground is ∆r TG = v TG ∆t. As measured by Greg, Wanda’s displacement is
Page 1 and 2: 3.5 Motion in a Plane with Constant
Page 3 and 4: 3.5 Motion in a Plane with Constant
Page 5: 3.5 Motion in a Plane with Constant

Motion of Projectiles

Create successful ePaper yourself

Delete template?

Save as template?