Projectile Motion Initial velocity is at an angle

Projectile Motion Initial velocity is at an angle with respect to the horizontal. The only force on the projectile is the downward gravitational force of Earth on the projectile.

Path

Example If you toss a ball horizontally in the +x direction, sketch the path of the projectile.

Example While standing on a 10 -m high-dive, you throw a ball downward at an angle of 30 from the horizontal. Sketch the path of the ball.

x-component of motion

y-component of motion

Velocity components vx and vy A ball is rolled horizontally off a table. Assume air resistance is negligible. Then between the instant the ball leaves the table and the instant it strikes the floor, _____ A. vx = 0 and vy is constant but ≠ 0. B. vx and vy are both constant but ≠ 0. C. vx is constant but ≠ 0 and vy is increasing in magnitude. D. vx is constant but ≠ 0 and vy is decreasing in magnitude. E. vx and vy are both increasing in magnitude.

Acceleration components ax and ay A ball is rolled horizontally off a table. Assume air resistance is negligible. Then between the instant the ball leaves the table and the instant it strikes the floor, _____ A. ax = 0 and ay is constant but ≠ 0. B. ax and ay are both constant but ≠ 0. C. ax is constant but ≠ 0 and ay is increasing in magnitude. D. ax is constant but ≠ 0 and ay is decreasing in magnitude. E. ax and ay are both increasing in magnitude.

Note: Each answer has a part for each blank. With negligible air resistance, the thrown ball follows the path traced in red. As it does so, the velocity of the shadow that it casts on the wall ____ and the velocity of the shadow that it casts on the ground _____. A. becomes more positive and then less positive … doesn’t change B. changes sign … doesn’t change C. becomes more positive and then less positive … becomes more positive and then less positive D. changes sign … becomes more positive and then less positive

2 + 1 3 +x With negligible air resistance, the thrown ball follows the path traced in red. In the coordinate frame shown, the ball’s acceleration is _____. A. negative at points 1, 2, and 3 B. negative at point 1, zero at point 2, and positive at point 3 C. positive at point 1, zero at point 2, and negative at point 3 D. positive at points 1 and 3, but zero at point 2 E. negative at points 1 and 3, but zero at point 2

v 0 q A test rocket, due to technical failure, experiences no further thrust from the instant it leaves its ground level launching pad until it lands on the ground some distance away. a. During its flight, the rocket’s horizontal position x _____ each time the clock advances by one second (Dt = 1 s). A. changes by the same positive amount B. changes by the same negative amount C. remains unchanged D. increases when the rocket is going up and decreases when it is coming back down

v 0 q A test rocket, due to technical failure, experiences no further thrust from the instant it leaves its ground level launching pad until it lands on the ground some distance away. b. During its flight, the rocket’s vertical position y _____ each time the clock advances by one second (Dt = 1 s). A. changes by the same positive amount B. changes by the same negative amount C. remains unchanged D. increases when the rocket is going up and decreases when it is coming back down

v 0 q A test rocket, due to technical failure, experiences no further thrust from the instant it leaves its ground level launching pad until it lands on the ground some distance away. c. During its flight, the rocket’s horizontal velocity component vx _____ each time the clock advances by one second (Dt = 1 s). A. changes by the same positive amount B. changes by the same negative amount C. remains unchanged D. increases when the rocket is going up and decreases when it is coming back down

v 0 q A test rocket, due to technical failure, experiences no further thrust from the instant it leaves its ground level launching pad until it lands on the ground some distance away. d. During its flight, the rocket’s vertical velocity component vy _____ each time the clock advances by one second (Dt = 1 s). A. changes by the same positive amount B. changes by the same negative amount C. remains unchanged D. increases when the rocket is going up and decreases when it is coming back down

v 0 q A test rocket, due to technical failure, experiences no further thrust from the instant it leaves its ground level launching pad until it lands on the ground some distance away. e. At the highest point in it’s trajectory, the rocket’s velocity _______. A. is zero B. is horizontal C. is vertical D. has both a horizontal and a vertical component

y 1. . 2 x The graph above shows the path of a projectile that is unaffected by air resistance. The values of _______ are both different at point 1 than at point 2. A. y and vy B. x and vx C. x and v D. x and vy E. ay and x

y 1. . 2 x The graph above shows the path of a projectile that is unaffected by air resistance. The quantities _______ both have the same values at point 1 that they do at point 2. A. y and vy B. x and vx C. ay and v D. ay and vy E. ax and x

Equations of Motion

Useful derived equations Range

Useful derived equations Peak height

Useful derived equations time of flight

Example A child swings on a playground swing and jumps off when the swing has a velocity of magnitude 5 m/s at an angle of 40 and she is 1 m off the ground. (a) What is her peak height and (b) where does she land?

Example The world record shot put is 23. 12 m by American Randy Barnes in 1990. If the shot left his hand at a height of 1. 5 m and an angle of 45 , what was the shot’s initial speed?