Do Now A tennis ball is dropped from the top of a building. It strikes the ground 6 seconds after being dropped. How high is the building? (b) What is the speed of the ball the instant it strikes the ground?

An object near the surface of planet X falls freely from rest and reaches a speed of 12.0 meters per second after it has fallen 14.4 meters. What is the acceleration due to gravity on planet X?

Do Now Anna drops a ball from rest from the top of 78.4 meter high cliff. How much time will it take for the ball to reach the ground?

Solution: Given: y = 78. 4m vy0 = 0 a = g = 9 Solution: Given: y = 78.4m vy0 = 0 a = g = 9.8m/s2 Find: t y = vy0t + 1/2 at2 78.4m = 0 + x (½)(9.8m/s2)t2 t = 4s

What is Projectile Motion?

Projectile Motion Two-dimensional motion of an object Vertical Horizontal

Horizontal Projectile Motion In a projectile motion, vertical motion and horizontal motion can be treated separately. Vertical – like free fall Horizontal – 0 acceleration (constant velocity)

Suppose a rescue airplane drops a relief package while it is moving with a constant horizontal speed at an elevated height. Assuming that air resistance is negligible, where will the relief package land relative to the plane? a. below the plane and behind it. b. directly below the plane c. below the plane and ahead of it

Projectile Motion It can be understood by analyzing the horizontal and vertical motions separately. Figure 3-20. Caption: Projectile motion of a small ball projected horizontally. The dashed black line represents the path of the object. The velocity vector at each point is in the direction of motion and thus is tangent to the path. The velocity vectors are green arrows, and velocity components are dashed. (A vertically falling object starting at the same point is shown at the left for comparison; vy is the same for the falling object and the projectile.)

Projectile Motion 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. Figure 3-21. Caption: Multiple-exposure photograph showing positions of two balls at equal time intervals. One ball was dropped from rest at the same time the other was projected horizontally outward. The vertical position of each ball is seen to be the same at each instant.

Sample problem 1 Fred throws a baseball 42m/s horizontally from a height of two meters. How far will the ball travel before it reaches the ground?

A movie stunt driver on a motorcycle speeds horizontally off a 50 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.

A cannonball is launched horizontally from the top of an 78 A cannonball is launched horizontally from the top of an 78.4-meter high cliff. How much time will it take for the ball to reach the ground?

A rock is thrown horizontally off a 100m cliff. It lands 95m away A rock is thrown horizontally off a 100m cliff. It lands 95m away. At what speed was it thrown?

Projectile A is launched horizontally at a speed of 20 meter per second from the top of a cliff and strikes a level surface below, 3.0 seconds later. Projectile B is launched horizontally fro the same location at a speed of 30 meters per second. The time it takes projectile B to reach the level surface is A) 4.5s B) 2.0s C) 3.0s D) 10S

The above information can be summarized by the following table. Horizontal Motion Vertical Forces (Present? - Yes or No) (If present, what dir'n?) No Yes The force of gravity acts downward Acceleration Yes "g" is downward at 9.8 m/s/s Velocity (Constant or Changing?) Constant Changing (by 9.8 m/s each second)

The cannonball launched by a cannon from the top of a very high cliff.

The cannonball launched by a cannon from the top of a very high cliff.

The cannonball launched by a cannon from the top of a very high cliff. and causes the parabolic trajectory which is characteristic of projectiles.   The cannonball launched by a cannon from the top of a very high cliff.

Do Now A cannonball is launched at 650m/s off a cliff 5m high. What is it final horizontal velocity?

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. Figure 3-26. Response: Both the water balloon and the boy in the tree start falling at the same instant, and in a time t they each fall the same vertical distance y = ½ gt2, much like Fig. 3–21. In the time it takes the water balloon to travel the horizontal distance d, the balloon will have the same y position as the falling boy. Splat. If the boy had stayed in the tree, he would have avoided the humiliation.

Do Now An arrow is released at 30 degrees above the horizontal at 50m/s. Find its horizontal and vertical component of initial velocity.

Ground (Angled) Projectiles For objects launched at an angle, we need to determine the initial velocity in both the horizontal and vertical directions first. Vx0 =V0 cosθ Vy0 =V0 sinθ

The time of flight of a projectile is twice the time to rise to the peak.

A cannonball is launched from level ground at an angle of 300 above the horizontal with an initial velocity of 26 m/s. How far does the cannonball travel horizontally before returning with the ground?

An arrow is released at 30 degrees above the horizontal at 50m/s and travels 750m. When does it land?

Class Exercise An object is fired from the ground at 100 meters per second at an angle of 30 degrees with the horizontal Calculate the horizontal and vertical components of the initial velocity After 2.0 seconds, how far has the object traveled in the horizontal direction? How high is the object at this point?

Solution Part a Part b Part c

Do Now Megan hits a golf ball with a velocity of 25 m/s at an angle of 60°to the horizontal. Answer following questions. a)What is the horizontal initial velocity? b)What is the vertical initial velocity? c)How long does it take for the ball reach its highest point? d)How long does it take for the ball return to the ground? e)Calculate the total horizontal distance traveled by the ball. f)What is the peak height of the ball?

a) Vx0 = v0 cosΘ = (25m/s) cos60°= 12.5 m/s Solution a) Vx0 = v0 cosΘ = (25m/s) cos60°= 12.5 m/s Vy0 = v0 sinΘ = (25m/s) sin60°= 21.7 m/s c) vy = vy0 + gtup vy = 0 g = - 9.8m/s2 0 = 21.7m/s + (-9.8m/s2) tup tup = 2.21s d) ttotal = 2tup = 2 x (2.21s) = 4.42s e) dx = vx0 t = (12.5m/s)(4.42s) = 55.2m f) h = vy0t +1/2· g·t2 = (21.7m/s) x (2.21s) + 1/2 (-9.8m/s2) x (2.21s)2 = 23.9m

Projectile Motion If an object is launched at an initial angle of θ0 with the horizontal, the analysis is similar except that the initial velocity has a vertical component. Figure 3-22. Caption: Path of a projectile fired with initial velocity v0 at angle θ0 to the horizontal. Path is shown dashed in black, the velocity vectors are green arrows, and velocity components are dashed. The acceleration a = dv/dt is downward. That is, a = g = -gj where j is the unit vector in the positive y direction.

The projectile is launched upward at an angle to the horizontal

Do Now A ball is kicked at an angle 30° to the horizontal and velocity of 30m/s. Answer following questions. a)What is the horizontal initial velocity? b)What is the vertical initial velocity? c)How long does it take for the ball reach its highest point? d)How long does it take for the ball return to the ground? e)Calculate the total horizontal distance traveled by the ball. f)What is the peak height of the ball?

Projectile Motion Any object that is moving through the air affected only by gravity is called a projectile. The path a projectile follows is called its trajectory. The projectile moves along a parabolic trajectory.

Range of projectile The distance a projectile travels horizontally is called its range. The range, dx increase when Ɵ increase from 0° to 45°. dx decrease when Ɵ increase from 45° to 90° . In other words, dx will reach the maximum when the angle is 45°.

PROJECTILE MOTION - SUMMARY Projectile motion is motion with a constant horizontal velocity combined with a constant vertical acceleration. The projectile moves along a parabolic trajectory.

Projectile Motion The trajectory of a thrown basketball follows a special type of arch-shaped curve called a parabola. The distance a projectile travels horizontally is called its range.

Conceptual notions about projectiles. A projectile is any object upon which the only force is gravity. Projectiles travel with a parabolic trajectory due to the influence of gravity. There are no horizontal forces acting upon projectiles and thus no horizontal acceleration. The horizontal velocity of a projectile is constant (a never changing in value). There is a vertical acceleration caused by gravity; its value is 9.8 m/s/s, down. The vertical velocity of a projectile changes by 9.8 m/s each second. The horizontal motion of a projectile is independent of its vertical motion.

Gravity-Free Environment

Gravity Environment

Solving Problems Involving Projectile Motion Read the problem carefully, and choose the object(s) you are going to analyze. Draw a diagram. Choose an origin and a coordinate system. 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. Examine the x and y motions separately.

The cannonball launched by a cannon from the top of a very high cliff. "an object in motion will ...". This is Newton's law of inertia.   The cannonball launched by a cannon from the top of a very high cliff.

The answer is the same as the “Do Now” The answer is the same as the “Do Now”. Remember: Perpendicular components of motion are independent of each other.

A football is kicked at an angle θ0 = 30° with a velocity of 20 A football is kicked at an angle θ0 = 30° 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.

Projectile Motion A projectile is an object moving in two dimensions under the influence of Earth's gravity; its path is a parabola. Figure 3-19. Caption: This strobe photograph of a ball making a series of bounces shows the characteristic “parabolic” path of projectile motion.

A projectile is any object which once projected or dropped continues in motion by its own inertia and is influenced only by the downward force of gravity.

Each component is independent of the other. Projectile Motion Most important, the horizontal component of motion for a projectile is completely independent of the vertical component of motion. Each component is independent of the other. Their combined effects produce the variety of curved paths that projectiles follow.

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? Figure 3-25. Response: The child throws the apple straight up from her own reference frame with initial velocity vy0 (Fig. 3–25a). But when viewed by someone on the ground, the apple also has an initial horizontal component of velocity equal to the speed of the wagon, vx0. Thus, to a person on the ground, the apple will follow the path of a projectile as shown in Fig. 3–25b. The apple experiences no horizontal acceleration, so vx0 will stay constant and equal to the speed of the wagon. As the apple follows its arc, the wagon will be directly under the apple at all times because they have the same horizontal velocity. When the apple comes down, it will drop right into the outstretched hand of the child. The answer is (b).