20/06/ Projectile Motion. 3-7 Projectile Motion. 3-7 Projectile Motion. 3-7 Projectile Motion
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1 3-7 A projectile is an object moving in two dimensions under the influence of Earth's gravity; its path is a parabola. 3-7 It can be understood by analyzing the horizontal and vertical motions separately. 3-7 The speed in the x-direction is constant; in the y- direction the object moves with constant acceleration g. This photograph shows two balls that start to fall at the same time. The one on the right has an initial speed in the x-direction. It can be seen that vertical positions of the two balls are identical at identical times, while the horizontal position of the yellow ball increases linearly. 3-7 If an object is launched at an initial angle of θ 0 with the horizontal, the analysis is similar except that the initial velocity has a vertical component. Projectile motion is motion with constant acceleration in two dimensions, where the acceleration is g and is down. 1. Read the problem carefully, and choose the object(s) you are going to analyze. 2. Draw a diagram. 3. Choose an origin and a coordinate system. 4. Decide on the time interval; this is the same in both directions, and includes only the time the object is moving with constant acceleration g. 5. Examine the x and y motions separately. 1
2 6. List known and unknown quantities. Remember that v x never changes, and that v y = 0 at the highest point. 7. Plan how you will proceed. Use the appropriate equations; you may have to combine some of them. Example 3-6: Driving off a cliff. A movie stunt driver on a motorcycle speeds horizontally off a 50.0-m-high cliff. How fast must the motorcycle leave the cliff top to land on level ground below, 90.0 m from the base of the cliff where the cameras are? Ignore air resistance. Example 3-7: A kicked football. A football is kicked at an angle θ 0 = 37.0 with a velocity of 20.0 m/s, as shown. Calculate (a) the maximum height, (b) the time of travel before the football hits the ground, (c) how far away it hits the ground, (d) the velocity vector at the maximum height, and (e) the acceleration vector at maximum height. Assume the ball leaves the foot at ground level, and ignore air resistance and rotation of the ball. 2
3 Conceptual Example 3-8: Where does the apple land? A child sits upright in a wagon which is moving to the right at constant speed as shown. The child extends her hand and throws an apple straight upward (from her own point of view), while the wagon continues to travel forward at constant speed. If air resistance is neglected, will the apple land (a) behind the wagon, (b) in the wagon, or (c) in front of the wagon? Conceptual Example 3-9: The wrong strategy. A boy on a small hill aims his water-balloon slingshot horizontally, straight at a second boy hanging from a tree branch a distance d away. At the instant the water balloon is released, the second boy lets go and falls from the tree, hoping to avoid being hit. Show that he made the wrong move. (He hadn t studied physics yet.) Ignore air resistance. Example 3-10: Level horizontal range. (a) Derive a formula for the horizontal range R of a projectile in terms of its initial speed v 0 and angle θ 0. The horizontal range is defined as the horizontal distance the projectile travels before returning to its original height (which is typically the ground); that is, y(final) = y 0. (b) (b) Suppose one of Napoleon s cannons had a muzzle speed, v 0, of 60.0 m/s. At what angle should it have been aimed (ignore air resistance) to strike a target 320 m away? 3
4 Example Suppose the football in Example 3 7 was punted and left the punter s foot at a height of 1.00 m above the ground. How far did the football travel before hitting the ground? Set x 0 = 0, y 0 = 0. Example 3-12: Rescue helicopter drops supplies. A rescue helicopter wants to drop a package of supplies to isolated mountain climbers on a rocky ridge 200 m below. If the helicopter is traveling horizontally with a speed of 70 m/s (250 km/h), (a) how far in advance of the recipients (horizontal distance) must the package be dropped? (b) Suppose, instead, that the helicopter releases the package a horizontal distance of 400 m in advance of the mountain climbers. What vertical velocity should the package be given (up or down) so that it arrives precisely at the climbers position? (c) With what speed does the package land in the latter case? 4
5 Projectile motion is parabolic: Taking the equations for x and y as a function of time, and combining them to eliminate t, we find y as a function of x: Examples of projectile motion. Notice the effects of air resistance. This is the equation for a parabola. 5
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