Chapter 4 Motion in Two Dimensions EXAMPLES

Example 4.1 Driving off a cliff. y i = 0 at top, y is positive upward. Also v yi = 0 How fast must the motorcycle leave the cliff to land at x = 90 m, y = – 50 m Unknown: v xi = ? Formulas: v y =  gt x = v xi t y =  ½gt 2 Time to Bottom: v xi = x/t = 90.0m/3.19s  v xi = 28.2 m/s

Example 4.2 Kicked football Given: θ i = 37º, v i = 20 m/s v xi = v i cosθ i = 16 m/s & v yi = v i sinθ i = 12 m/s Find a. Max height (h) ? b. Time when hits ground? c. Total distance traveled in the x direction (R) ? d. Velocity at top? e. Acceleration at top? v yi v xi vivi

Example 4.2 cont.

Example 4.3 Where Does The Apple Land? A child sits in a wagon, moving to the right (x-direction) at constant velocity v ox. She throws an apple straight up (from her viewpoint) with initial velocity v oy while she continues to travel forward at v ox Neglect air resistance. Will the apple land behind the wagon, in front of the wagon, or in the wagon?

Example 4.3 Cont. The apple will stay above the girl the entire trip and will land in the wagon. The reason is: To a person in the ground reference frame (b) the apple will be exactly a projectile in motion (neglecting air resistance). To the girl it is an object in free fall. And the Vertical motion of a projectile and free fall are the same.

Example 4.4 Wrong Strategy (Similar to Example 4.3 Text Book) “Shooting the Monkey”!! 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 an fall from the tree, hopping to avoid being hit. Show that he made the wrong move (He hadn’t studied Physics yet!!)

Example 4.4 Cont. “ Shooting the Monkey”!! Both the water balloon and the boy in the tree start falling at the same time, and in a time t they each fall the same vertical distance y = ½gt 2 In the same time it takes the water balloon to travel the horizontal distance d, the balloon will have the same y position as the falling boy. Splash!!! If the boy had stayed in the tree, he would have avoided the humiliation

Example 4.5 That’s Quite an Arm Non-Symmetric Projectile Motion Example 4.4 (text book), page 84 Follow the general rules for projectile motion Break the y-direction into parts up and down or symmetrical back to initial height and then the rest of the height May be non-symmetric in other ways

Example 4.5 Cont. Given: θ i = 30º, v i = 20 m/s  (A) v xi = v i cosθ i = 17.3 m/s and v yi = v i sinθ i = 10.0 m/s At t = 0 : x i = 0 y i = 0 Find: t = ? (time at which the stone hits the ground) with y f = – 45.0m Using: y f = v yi t – ½gt 2  – 45.0m = (10.0m/s)t – (4.90m/s 2 )t 2 Solving for t using General Quadratic Formula: t = 4.22 s (B) v xi = v xf = 17.3 m/s and v yf = v yi – gt  v yf = 10.0m/s – (9.80m/s 2 )(4.22s)  v yf = ̶ 31.4m/s 

Example 4.6 The End of the Ski Jump Example 4.5 (text book), page 85 Given the figure of the ski jumper, find the distance d traveled along the incline. 1. Coordinates x and y at the end: 2. From the figure:

Example 4.6 Cont. 3. Equating (1) = (3) and (2) = (4) 4. Dividing (6) by (5): 5. Substitution of (7) in (5) and solving for d: 6. Substitution of d into (3) and (4), gives the coordinates:

Example 4.7 The Centripetal Acceleration of the Earth Calculate a c of the Earth, assuming it moves in a circular orbit around the Sun. Note that a c << g

Material from the book to Study!!! Objective Questions: Conceptual Questions: Problems: Material for the Midterm