Motion of Projectiles

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66 Chapter 3 Motion in a Plane y v y v y v y = 0 Figure 3.17 Motion diagram showing the trajectory of a projectile. The position is drawn at equal time intervals. Superimposed are the velocity vectors along with their x- and y-components. v iy v ix v v ix ix v v y ix v ix θ v y v ix v ix v fy x The horizontal and vertical motions of a projectile can be treated separately; they are independent of each other. With the y-axis pointing up, a y = –g because the acceleration is downward (in the –y direction). The acceleration has no x-component (a x = 0), so the stone’s horizontal velocity component v x is constant. The vertical velocity component v y changes at a constant rate, exactly as if the stone were propelled straight up with an initial speed of v iy . The initially positive v y decreases until, at the top of flight, v y = 0. Then the pull of gravity makes the projectile fall back downward. During the downward trip, v y is still changing at the same constant rate with which it changed on the way up and at the top of the path. The acceleration has the same constant value—magnitude and direction—for the entire path. The displacement of the projectile at any instant is the vector sum of the displacements in the two mutually perpendicular directions. The motion of a projectile when air resistance is negligible is the superposition of horizontal motion with constant velocity and vertical motion with constant acceleration. The vertical and horizontal motions each proceed independently, as if the other motion were not present. In the experiment of Fig. 3.18, one ball was dropped and, at the same instant, another was projected horizontally. The strobe photo shows snapshots of the two balls at equally spaced time intervals. The vertical motion of the two is identical; at every instant, the two are at the same height. The fact that they have different horizontal motion does not affect their vertical motion. (This statement would not be true if air resistance were significant.) Figure 3.18 Independence of horizontal and vertical motion of a projectile in the absence of air resistance. The vertical motion of the projectile (white) is the same as that of an object (red) that falls straight down. PHYSICS AT HOME Take a nickel and a penny to a room with a high table or countertop. Place the penny at the edge of the table and then slide the nickel so it collides with the penny. Listen for the sound of the two coins hitting the floor. The two coins will slide off the table with different horizontal velocities but will land at the same time.
3.5 Motion in a Plane with Constant Acceleration 67 Conceptual Example 3.4 Trajectory of a Projectile The graph of an equation of the form y = kx 2 , k = a nonzero constant is a parabola. Show that the trajectory of a projectile is a parabola. [Hint: Choose the origin at the highest point of the trajectory and let t i = 0 at that instant.] Strategy and Solution We start at the high point of the path and look at displacements from there. The horizontal displacement is proportional to the elapsed time t since the horizontal velocity is constant. The vertical displacement is the average vertical velocity component times the elapsed time t. The average vertical velocity component is itself proportional to t since it changes at a constant rate. Therefore, the vertical displacement is proportional to t 2 . Thus, the vertical displacement y is proportional to the square of the horizontal displacement x and y = kx 2 , where k is a constant of proportionality. The path followed by a projectile in free fall is a parabola. Discussion The same conclusion can be drawn algebraically. With the +y-axis upward and the origin and t = 0 at the top of flight, x i , y i , and v iy are all zero. Then x = v ix t and y = v iy t + 1 2 a yt 2 = – 1 2 gt 2 = – 1 2 x g g v x 2 2 ix = – 2v So y is proportional to x 2 and the constant of proportionality is –g/(2v 2 ix). Conceptual Practice Problem 3.4 Stones 2 ix Throwing You stand at the edge of a cliff and throw stones horizontally into the river below. To double the horizontal displacement of a stone from the cliff to where it lands, by what factor must you increase the stone’s initial speed? Neglect air resistance. Figure 3.19 shows graphs of the x- and y-components of the velocity and position of a projectile as functions of time. In this case, the projectile is launched above flat ground at t = 0 and returns to the same elevation at a later time t f . Note that the y-component graphs are symmetrical about the vertical line through the highest point in the trajectory. The y-component of velocity decreases linearly from its initial value; the slope of the line is a y = –g. When v y = 0, the projectile is at the apex of its trajectory. Then v y continues to decrease at the same rate and is now negative with its magnitude getting larger and larger. At t f , when the projectile has returned to its original altitude, the y-component of the velocity has the same magnitude as at t = 0 but with the opposite sign (v y = –v iy ). The graph of y(t) indicates that the projectile moves upward, quickly at first and then gradually slowing, until it reaches the maximum height. The slope of the tangent to the y(t) graph at any particular moment of time is v y at that instant. At the highest point of the y(t) graph, the tangent is horizontal and v y = 0. After that, gravity makes the projectile start to fall downward. The horizontal velocity is constant, so the graph of v x (t) is a horizontal line. The horizontal position x increases uniformly in time because the object is moving with a constant v x . Vertical velocity v y v iy 0 v fy 0 1 – 2t f t f Time Horizontal velocity v x v ix 0 0 1 – 2t f t f Time Vertical position y 0 0 1 – 2t f t f Time Horizontal position x 0 0 1 – 2t f t f Time Figure 3.19 Projectile motion: separate vertical and horizontal quantities versus time.
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Page 7: 3.6 Velocity Is Relative; Reference

Motion of Projectiles

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