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7.2 Projectile Motion and the Velocity Vector

7.2 Projectile Motion and the Velocity Vector. Chapter 7 Objectives. Add and subtract displacement vectors to describe changes in position. Calculate the x and y components of a displacement, velocity, and force vector. Write a velocity vector in polar and x-y coordinates.

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7.2 Projectile Motion and the Velocity Vector

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  1. 7.2 Projectile Motion and the Velocity Vector

  2. Chapter 7 Objectives Add and subtract displacement vectors to describe changes in position. Calculate the x and y components of a displacement, velocity, and force vector. Write a velocity vector in polar and x-y coordinates. Calculate the range of a projectile given the initial velocity vector. Use force vectors to solve two-dimensional equilibrium problems with up to three forces. Calculate the acceleration on an inclined plane when given the angle of incline.

  3. Chapter 7 Vocabulary • Cartesian coordinates • component • cosine • displacement • inclined plane • magnitude • parabola • polar coordinates • projectile • Pythagorean theorem • range • resolution • resultant • right triangle • scalar • scale • sine • tangent • trajectory • velocity vector • x-component • y-component

  4. Inv 7.2 Projectile Motion Investigation Key Question: How can you predict the range of a launched marble?

  5. 7.2 Projectile Motion and the Velocity Vector 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.

  6. 7.2 Projectile Motion and the Velocity Vector 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.

  7. 7.2 The velocity vector The velocity vector (v) is a way to precisely describe the speed and direction of motion. There are two ways to represent velocity. Both tell how fast and in what direction the ball travels.

  8. Draw the velocity vector v = (5, 5) m/sec and calculate the magnitude of the velocity (the speed), using the Pythagorean theorem. Drawing a velocity vectorto calculate speed • You are asked to sketch a velocity vector and calculate its speed. • You are given the x-y component form of the velocity. • Set a scale of 1 cm = 1 m/s. Draw the sketch. Measure the resulting line segment or use the Pythagorean theorem: a2 + b2 = c2 • Solve: v2 = (5 m/s)2 + (5 m/s)2 = 50 m2 /s2 • v = 0 m2 /s2 = 7.07 m/s

  9. 7.2 The components of the velocity vector Suppose a car is driving 20 meters per second. The direction of the vector is 127 degrees. The polar representation of the velocity is v = (20 m/sec, 127°).

  10. Calculating the componentsof a velocity vector • You are asked to calculate the components of the velocity vector. • You are given the initial speed and angle. • Draw a diagram to scale or use vx= v cos θ and vy= v sin θ. • Solve: • vx = (10 m/s)(cos 30o) = (10 m/s)(0.87) = 8.7 m/s • vy= (10 m/s)(sin 30o) = (10 m/s)(0.5) = 5 m/s A soccer ball is kicked at a speed of 10 m/s and an angle of 30 degrees. Find the horizontal and vertical components of the ball’s initial velocity.

  11. 7.2 Adding velocity vectors Sometimes the total velocity of an object is a combination of velocities. • One example is the motion of a boat on a river. • The boat moves with a certain velocity relative to the water. • The water is also moving with another velocity relative to the land.

  12. 7.2 Adding Velocity Components • Velocity vectors are added by components, just like displacement vectors. • To calculate a resultant velocity, add the x components and the y components separately.

  13. Calculating the componentsof a velocity vector • You are asked to calculate the resultant velocity vector. • You are given the plane’s velocity and the wind velocity • Draw diagrams, use Pythagorean theorem. • Solve and add the components to get the resultant velocity : • Plane: vx= 100 cos 30o = 86.6 m/s, vy= 100 sin 30o = 50 m/s • Wind: vx= 40 cos 45o = 28.3 m/s, vy= - 40 sin 45o = -28.3 m/s • v = (86.6 + 28.3, 50 – 28.3) = (114.9, 21.7) m/s or (115, 22) m/s An airplane is moving at a velocity of 100 m/s in a direction 30 degrees northeast relative to the air. The wind is blowing 40 m/s in a direction 45 degrees southeast relative to the ground. Find the resultant velocity of the airplane relative to the ground.

  14. 7.2 Projectile motion When we drop a ball from a height we know that its speed increases as it falls. The increase in speed is due to the acceleration gravity, g = 9.8 m/sec2. Vx Vy y x

  15. 7.2 Horizontal motion The ball’s horizontal velocity remains constant while it falls because gravity does not exert any horizontal force. Since there is no force, the horizontal acceleration is zero (ax = 0). The ball will keep moving to the right at 5 m/sec.

  16. 7.2 Horizontal motion The horizontal distance a projectile moves can be calculated according to the formula:

  17. 7.2 Vertical motion The vertical speed (vy) of the ball will increase by 9.8 m/sec after each second. After one second has passed, vyof the ball will be 9.8 m/sec. After 2 seconds have passed, vywill be 19.6 m/sec and so on.

  18. Analyzing a projectile A stunt driver steers a car off a cliff at a speed of 20 meters per second. He lands in the lake below two seconds later. Find the height of the cliff and the horizontal distance the car travels. • You are asked for the vertical and horizontal distances. • You know the initial speed and the time. • Use relationships: y = voyt – ½ gt2 and x = vox t • The car goes off the cliff horizontally, so assume voy= 0. Solve: • y = – (1/2)(9.8 m/s2)(2 s)2y = –19.6 m. (negative means the car is below its starting point) • Use x = voxt, to find the horizontal distance: x = (20 m/s)(2 s) x = 40 m.

  19. 7.2 Projectiles launched at an angle A soccer ball kicked off the ground is also a projectile, but it starts with an initial velocity that has both vertical and horizontal components. *The launch angle determines how the initial velocity divides between vertical (y) and horizontal (x) directions.

  20. 7.2 Steep Angle A ball launched at a steep angle will have a large vertical velocity component and a small horizontal velocity.

  21. 7.2 Shallow Angle A ball launched at a low angle will have a large horizontal velocity component and a small vertical one.

  22. 7.2 Projectiles Launched at an Angle The initial velocity components of an object launched at a velocity vo and angle θ are found by breaking the velocity into x and y components.

  23. 7.2 Range of a Projectile The range, or horizontal distance, traveled by a projectile depends on the launch speed and the launch angle.

  24. 7.2 Range of a Projectile The range of a projectile is calculated from the horizontal velocity and the time of flight.

  25. 7.2 Range of a Projectile A projectile travels farthest when launched at 45 degrees.

  26. 7.2 Range of a Projectile The vertical velocity is responsible for giving the projectile its "hang" time.

  27. 7.2 "Hang Time" You can easily calculate your own hang time. Run toward a doorway and jump as high as you can, touching the wall or door frame. Have someone watch to see exactly how high you reach. Measure this distance with a meter stick. The vertical distance formula can be rearranged to solve for time:

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