Chapter Projectile Motion 6.1

Projectile Motion 6.1 In this section you will: Recognize that the vertical and horizontal motions of a projectile are independent. Relate the height, time in the air, and initial vertical velocity of a projectile using its vertical motion, and then determine the range using the horizontal motion. Explain how the trajectory of a projectile depends upon the frame of reference from which it is observed. Read Chapter 6.1. HW 6.A: Handout Projectile Motion Study Guide, due before Chapter Test.

Projectile Motion 6.1 Projectile Motion Section Projectile Motion 6.1 Projectile Motion If you observed the movement of a golf ball being hit from a tee, a frog hopping, or a free throw being shot with a basketball, you would notice that all of these objects move through the air along similar paths, as do baseballs, arrows, and bullets. Each path is a curve that moves upward for a distance, and then, after a time, turns and moves downward for some distance. You may be familiar with this curve, called a parabola , from math class.

Section Projectile Motion 6.1 An object shot through the air is called a projectile . A projectile can be a football, a bullet, or a drop of water. You can draw a free-body diagram of a launched projectile and identify all the forces that are acting on it. No matter what the object is, after a projectile has been given an initial thrust, if you ignore air resistance, it moves through the air only under the force of gravity . The force of gravity is what causes the object to curve downward in a parabolic flight path. Its path through space is called its trajectory . Demonstration & Activity: Horizontal Projectile Motion

Click image to view movie. Section Projectile Motion 6.1 Independence of Motion in Two Dimensions Click image to view movie. movanim 6.1

Projectile Motion 6.1 Trajectories Depend upon the Viewer Section Projectile Motion 6.1 Trajectories Depend upon the Viewer The path of the projectile, or its trajectory, depends upon who is viewing it. Suppose you toss a ball up and catch it while riding in a bus. To you, the ball would seem to go straight up and straight down. But an observer on the sidewalk would see the ball leave your hand, rise up, and return to your hand, but because the bus would be moving, your hand also would be moving. The bus, your hand, and the ball would all have the same horizontal velocity.

Section Projectile Motion 6.1 All objects, when ignoring air resistance, fall with the same acceleration, g = 9.8 m/s2 downward. The distance the ball falls each second increases because the ball is accelerating downward. The velocity also increases in the downward direction as the ball drops. This is shown by drawing a longer vector arrow for each time interval.

Section Projectile Motion 6.1 Vectors can also be used to represent a ball rolling horizontally on a table at a constant velocity. Newton’s 1st Law tells us the ball will continue rolling in a straight line at constant velocity unless acted on by an outside force. Each vector arrow is drawn the same length to represent the constant velocity. The velocity would remain constant but in the real world, friction makes it slow down and eventually stop.

Section Projectile Motion 6.1 Now, combine the motion of the ball in free fall with the motion of the ball rolling on the table at a constant velocity. This is seen when rolling the ball off of the table. The ball rolling on the table would continue forever in a straight line if gravity is ignored. The ball in free fall would continue to increase its speed if air resistance is ignored.

Section Projectile Motion 6.1 Since the ball is moving at a constant velocity and in free fall at the same time, the horizontal and vertical vectors are added together during equal time intervals. This is done for each time interval until the ball hits the ground. The path the ball follows can be seen by connecting the resultant vectors.

Projectile Motion 6.1 Look at the components of the velocity vectors. Section Projectile Motion 6.1 Look at the components of the velocity vectors. The length of the horizontal component stays the same for the whole time. The length of the vertical component increases with time. How do we combine the horizontal and vertical components to find the velocity vector?

Projectiles Launched at an Angle 6.1 Section Projectiles Launched at an Angle 6.1 Demonstration: Tossing a Ball If the object is launched upward, like a ball tossed straight up in the air, it rises with slowing speed, reaches the top of its path, and descends with increasing speed. A projectile launched at an angle would continue in a straight line at a constant velocity if gravity is ignored. However, gravity makes the projectile accelerate to Earth. Notice the projectile follows a parabolic trajectory.

Projectiles Launched at an Angle 6.1 Section Projectiles Launched at an Angle 6.1 Since the projectile is launched at an angle, it now has both horizontal and vertical velocities. The horizontal component of the velocity remains constant. The vertical component of the velocity changes as the projectile moves up or down.

Projectiles Launched at an Angle 6.1 Section Projectiles Launched at an Angle 6.1 The up and right vectors represent the velocity given to the projectile when launched. The vertical vectors decrease in magnitude due to gravity. Eventually, the effects of gravity will reduce the upward velocity to zero. This occurs at the top of the parabolic trajectory where there is only horizontal motion.

Projectiles Launched at an Angle 6.1 Section Projectiles Launched at an Angle 6.1 At the maximum height , the y component of velocity is zero. The x component remains constant. After gravity reduces the upward (vertical) speed to zero it begins to add a downward velocity. This velocity increases until the projectile return to the ground.

Projectiles Launched at an Angle 6.1 Section Projectiles Launched at an Angle 6.1 When looking at each half of the trajectory (up and down) you can determine that the speed of the projectile going up is equal to the speed of the projectile coming down (provided air resistance is ignored). The only difference is the direction of the motion. The other quantity depicted is the range which is the horizontal distance that the projectile travels. Not shown is the flight time, which is how much time the projectile is in the air. For football punts, flight time often is called hang time. range

Projectiles Launched at an Angle 6.1 Section Projectiles Launched at an Angle 6.1 Notice the x and y components of the velocity vector as the golf ball travels along its parabolic path.

Projectiles Launched at an Angle 6.1 Section Projectiles Launched at an Angle 6.1 Maximum range is achieved with a projection angle of 45° . For projection angles above and below 45°, the range is shorter, and it is equal for angles equally different from 45° (for example, 30° and 60°).

Section Projectile Motion 6.1 So far, air resistance has been ignored in the analysis of projectile motion. While the effects of air resistance are very small for some projectiles, for others, the effects are large and complex. For example, dimples on a golf ball reduce air resistance and maximize its range. The force due to air resistance does exist and it can be important.

Section Check 6.1 Question 1 A boy standing on a balcony drops a rock and throws another with an initial horizontal velocity of 3 m/s. Which of the following statements about the horizontal and vertical motions of the rocks are correct? (Neglect air resistance.) The rocks fall with a constant vertical velocity and a constant horizontal acceleration. The rocks fall with a constant vertical velocity as well as a constant horizontal velocity. The rocks fall with a constant vertical acceleration and a constant horizontal velocity. The rocks fall with a constant vertical acceleration and an increasing horizontal velocity.

Section Check 6.1 Answer 1 Answer: C Reason: The vertical and horizontal motions of a projectile are independent. The only force acting on the two rocks is force due to gravity. Because it acts in the vertical direction, the balls accelerate in the vertical direction. The horizontal velocity remains constant throughout the flight of the rocks.

Section Check 6.1 Question 2 Which of the following conditions is met when a projectile reaches its maximum height? Vertical component of the velocity is zero. Vertical component of the velocity is maximum. Horizontal component of the velocity is maximum. Acceleration in the vertical direction is zero.

Section Check 6.1 Answer 2 Answer: A Reason: The maximum height is the height at which the object stops its upward motion and starts falling down, i.e. when the vertical component of the velocity becomes zero.

Section Check 6.1 Question 3 Suppose you toss a ball up and catch it while riding in a bus. Why does the ball fall in your hands rather than falling at the place where you tossed it?

Section Section Check 6.1 Answer 3 Trajectory depends on the frame of reference. For an observer on the ground, when the bus is moving, your hand is also moving with the same velocity as the bus, i.e. the bus, your hand, and the ball will have the same horizontal velocity. Therefore, the ball will follow a trajectory and fall back in your hands.

Problem Solving with Projectile Motion Section 6.1 Problem Solving with Projectile Motion Problem Solving Strategy Sketch the problem. List givens and unknowns. Divide the projectile motion into a vertical motion problem and a horizontal motion problem. The vertical motion of a projectile is exactly that of an object dropped or thrown straight up or down with constant acceleration g. Use your constant acceleration (kinematics) equations. The horizontal motion of a projectile is the same as solving a constant velocity problem. Use dx = vxt and vxi = vxf. Vertical and horizontal motion are connected through the variable time .

Section Projectile Motion 6.1 Practice Problems, p. 150. 1 – 3. HW 6.A

Physics Chapter 6 Test Information Section 6.1 Physics Chapter 6 Test Information The test is worth 45 points total. Multiple Choice: 7 questions, 1 point each Problem Solving: 28 points Short Answer: 10 points

Projectile Motion Review Section 6.1 Projectile Motion Review Formulas: dy = vit + ½ a t2 constant acceleration in the y- direction dx = vx t constant velocity in the x- direction t = 2dy for a projectile that is launched horizontally, g the time only depends on the height Key Point: In projectile motion, the vertical and horizontal components of motion are independent.