What is Projectile Motion?

Instructional Objectives: Students will be able to: –Define Projectile Motion –Distinguish between the different types of projectile motion –Apply the concept to a ball rolling off a table and measure its velocity

Projectile Motion Same requirements as free fall, but projectile motion is two-dimensional motion of an object –V–Vertical direction –H–Horizontal direction

Concepts of Projectile Motion Horizontal –Motion of a ball rolling freely along a level surface –Horizontal velocity is ALWAYS constant –Nothing pushes or pulls on it in the horizontal direction to cause it to accelerate Vertical –Motion of a freely falling object –Force due to gravity –Vertical component of velocity changes with time Parabolic –Path traced by an object accelerating only in the vertical direction while moving at constant horizontal velocity

Parabola

Examples of Projectile Motion Launching a Cannon ball

Factors Affecting Projectile Motion What two factors would affect projectile motion? –Angle –Initial velocity Initial Velocity Angle

The range R is the horizontal distance the projectile has traveled when it returns to its launch height.

is the same for the horizontal and vertical components of the motion. Find from one component, then use that value for the other component. ASSESS Check that your result has the correct units, is reasonable, and answers the question. Projectile Motion Solutions PROBLEM-SOLVING STRATEGY Projectile motion problems MODEL Make simplifying assumptions. VISUALIZE Use a pictorial representation. Establish a coordinate system with the x-axis horizontal and the y- axis vertical. Show important points in the motion on a sketch. Define symbols and identify what the problem is trying to find. SOLVE The acceleration is known: and thus the problem becomes one of kinematics. The kinematic equations are (THESE ARE NOT NEW – THEY ARE JUST OUR 1-D EQUATIONS):

Horizontal motion No acceleration

Vertical motion (Equations of Motion ): 1 ) 2 ) 3 )

For a particular range less than the maximum and for a particular launch velocity, two different launch angles will give that range. The two angles add to give gives the maximum range.

Example Problem 1 An object is fired from the ground at 100 meters per second at an angle of 30 degrees with the horizontal a)Calculate the horizontal and vertical components of the initial velocity b)How long does it take to reach highest point? c)How far does the object travel in the horizontal direction?

A baseball is hit so that it leaves the bat making a 30 0 angle with the ground. It crosses a low fence at the boundary of the ballpark 100 m from home plate at the same height that it was struck. (Neglect air resistance.) What was its velocity as it left the bat? Example: A Home Run

Sample Problem 2 In Fig. 4-15, a rescue plane flies at 198 km/h (= 55.0 m/s) and a constant elevation of 500 m toward a point directly over a boating accident victim struggling in the water. The pilot wants to release a rescue capsule so that it hits the water very close to the victim.

(a) What should be the angle of the pilot's line of sight to the victim when the release is made? Solving for t, we find t = ± 10.1 s (take the positive root). Solution

(b) As the capsule reaches the water, what is its velocity as a magnitude and an angle? When the capsule reaches the water, 55 m/s 99 m/s 113 m/s 61°

Sample Problem 3 Figure 4-17 illustrates the flight of Emanuel Zacchini over three Ferris wheels, located as shown and each 18 m high. Zacchini is launched with speed v 0 = 26.5 m/s, at an angle = 53° up from the horizontal and with an initial height of 3.0 m above the ground. The net in which he is to land is at the same height. (a) Does he clear the first Ferris wheel?

SOLUTION The equation of trajectory when x 0 = 0 and y 0 = 0 is given by : Solving for y when x = 23m gives Since he begins 3m off the ground, he clears the Ferris wheel by (23.3 – 18) = 5.3 m

(b) If he reaches his maximum height when he is over the middle Ferris wheel, what is his clearance above it? SOLUTION: At maximum height, v y is 0. Therefore, and he clears the middle Ferris wheel by ( ) m =7.9 m

(c) How far from the cannon should the center of the net be positioned? SOLUTION:

FAST-MOVING PROJECTILES - SATELLITES Let’s throw stones horizontally with ever increasing velocity. The Earth’s curvature is 16 ft for every 5 miles (4.9 m for 8 km).

16 ft Curvature of Earth 5 miles Throw an objectfaster,faster.faster, We draw in each trajectory for 1 second

Ellipse Ellipse - Circle Ellipse Parabola V circle = 5 mi/s = 8 km/s V escape = 7 mi/s = 11 km/s Hyperbola

A rock is thrown upward at an angle. What happens to the horizontal component of its velocity as it rises? (Neglect air resistance.) (a) it decreases (b) it increases (c) it remains the same

In baseball which path would a home run most closely approximate? (Neglect air resistance.) (a) hyperbolic (b) parabolic (c) ellipse

A projectile is launched upward at an angle of 75 ° from the horizontal and strikes the ground a certain distance down range. What other angle of launch at the same launch speed would produce the same distance? (Neglect air resistance.) (a) 45 ° (b) 15 ° (c) 25 °

A horizontally traveling car drives off of a cliff next to the ocean. At the same time that the car leaves the cliff a bystander drops his camera. Which hits the ocean first? (Neglect air resistance.) (a) car (b) camera (c) they both hit at the same time