Introduction to Projectile Motion Projectile calculations ball rolling

Introduction to Projectile Motion ½ Projectile calculations (ball rolling off a table) Identifying x and y components Calculating time in air, height and range

Concept: Falling Bodies Y-component g=-9. 8 m/s 2 Formula: Δx = 1/2 gt 2 • A bullet shot horizontally and a bullet dropped will hit the ground at the same time. • Once released all bodies fall with the same acceleration vertically (y-component), on the surface of the Earth that is about -9. 8 m/s 2. • The horizontal or x-component has no acceleration. A ball dropped vertically and a ball thrown horizontally will land after the same amount of time. X-component No Acceleration Formula: vx=dx/t

Concept: Horizontal and Vertical Components • Also called x and y components • X component does not have acceleration – X Equation: v=d/t or with angle cosθ*vr=d/t – X Distance is sometimes called the range – X is called the horizontal component • Y component is the height and is accelerated motion of -9. 8 m/s 2 on Earth surface. – Also called the Vertical Component – Y equations use the Kinematics Equations – Y velocity with angle: sinθ*vr

Concept: Projectile Variables • Time is the same for both x and y components. • The velocity on the y changes both direction and magnitude during the flight. (negative ascending and positive descending). • The velocity on the x never changes. • The acceleration never changes magnitude or velocity during the entire flight. • At the apex the y acceleration is still 9. 8 m/s 2 and the y velocity is zero.

Concept: Importance of Time • Time is independent and the same for both the x and y components. • With the time both the range and the height can be calculated. • Calculating time from height or height from time can be accomplished with the following:

½ Projectile – Vertical Component Demonstration Problem: The Royal Gorge Bridge in Colorado rises 321 m above the Arkansas River. Suppose you kick a little rock horizontally off the bridge. The rock hits the water such that the magnitude of its horizontal displacement is 45. 0 m. Find the speed at which the rock was kicked. • Initial velocity on the y is 0. • Find the time by using the free-fall equation. t=sqrt(2*-321 m/-9. 8 m/s*s)=8. 1 sec • Find initial velocity on the x with time. vx = 45 m/8. 1 s = 5. 6 m/s • Extra - Find the final velocity on the y with time. vfy = 9. 8 m/s*s * 8. 1 s = 79. 4 m/s

½ Projectile – Finding Y from X Demonstration Problem: A cannon was shot horizontally off a cliff with a velocity of 20 m/s. If the ball lands 100 m from the base of the cliff what is the height of the cliff? • Given range and initial horizontal velocity. • Calculate height and time. • Find time first: t=dx/vx t 1/2 = 100 m/20 m/s = 5 sec Find height next h=(9. 8 m/s*s * 5 s 2)/2 = 122. 5 m

½ Projectile – Dropping from a Plane Demonstration Problem: A plane traveling horizontally at 100. 0 m/s at a height of 50. 0 m above the ground drops a package. What horizontal distance does the package travel before striking the ground? • Initial velocity on the y is 0. • Find the time by using the free-fall equation. t=sqrt(2*-50 m/-9. 8 m/s*s)= 3. 19 sec • Find the range on the x with time by rearranging v=d/t to solve for d. d=v*t dx = 100 m/s * 3. 19 s = 319 meters

Practice

Practice • During a thunderstorm, a tornado lifts a car to a height of 125 m above the ground. Increasing in strength, the tornado flings the car horizontally with an initial speed of 90. 0 m/s. How long does the car take to reach the ground? How far horizontally does the car travel before hitting the ground?

Answer Givens 1. Write the formula to find the vertical height 2. Find the time 3. Find the horizontal distance travelled

Practice 2 • A person standing at the edge of a seaside cliff kicks a stone over the edge with a speed of 18 m/s. The cliff is 52 m above the water’s surface, • How long does it take for the stone to fall to the water? With what speed does it strike the water?

Answer Problem #2 Givens Find the time

References • Holt Physics 2006