The centripetal acceleration for a particle moving in a circle is a c = v 2 /r, where v is its speed and r is its instantaneous radius of rotation.

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1 skiladæmi 1 Due: 11:59pm on Wednesday, September 9, 2015 You will receive no credit for items you complete after the assignment is due. Grading Policy Problem 3.04 The horizontal coordinates of a in a strong wind are given by and where and y are in meters, and t is in seconds. (a) What is the acceleration of the Frisbee? Give a magnitude m/s 2 Frisbee tm x 12t +4t 2 y 10t 3t 2, x Part B (b) Give a direction, measuring angles from the positive x direction. 323 Part C (c) What is the magnitude of the velocity at t 2.0 s, accurate to the nearest m/s? 4.00 m/s Alternative Exercise 3.91 On your first day at work for an appliance manufacturer, you are told to figure out what to do to the period of rotation during a washer spin cycle to triple the centripetal acceleration. You impress your boss by answering immediately. What do you tell her? Express your answer in terms of T.

2 T T Circular Launch A ball is launched up a semicircular chute in such a way that at the top of the chute, just before it goes into free fall, the ball has a centripetal acceleration of magnitude 2 g. How far from the bottom of the chute does the ball land? Your answer for the distance the ball travels from the end of the chute should contain R. Hint 1. Speed of ball upon leaving chute How fast is the ball moving at the top of the chute? Hint 1. Equation of motion The centripetal acceleration for a particle moving in a circle is a c v 2 /r, where v is its speed and r is its instantaneous radius of rotation. v (2gR) Hint 2. Time of free fall How long is the ball in free fall before it hits the ground? Express the free fall time in terms of R and g. Hint 1. Equation of motion There is constant acceleration due to gravity, so you can use the general expression

3 ayt 2. Write the values of y 0, vy0, and a y (separated by commas) that are appropriate for this situation. Use the standard convention that g is the magnitude of the acceleration due to gravity. Take y 0 at the ground, and take the positive y direction to be upward. y(t) + t+ y 0 v y0 2 y 0, vy0, a y 2R, 0, g Hint 2. Equation for the height of the ball To find the time in free fall before the ball hits the ground, tf, set the general equation for the height equal to the height of the ground. Answer in terms of tf, R, and g. y( ) 0 tf 2R g 2 t 2 f tf 2 R g Hint 3. Finding the horizontal distance D x( tf ) x0 + v x0tf 0 The horizontal distance follows from, where x0. vx0 and tf were found in Hints 1 and 2 respectively. D 2 2R Problem 3.18 A projectile is fired from point 0 at the edge of a cliff, with initial velocity components of v 0x 60.0 m/s and v 0y 175 m/s, as shown in the figure. The projectile rises and then falls into the sea at point P. The time of flight of the projectile is 40.0 s, and it experiences no appreciable air resistance in flight. What is the height of the cliff?

4 840 m Shooting over a Hill A projectile is fired with speed v0 at an angle θ from the horizontal as shown in the figure. Find the highest point in the trajectory, H. Express the highest point in terms of the magnitude of the acceleration due to gravity g, the initial velocity v0, and the angle θ. Hint 1. Velocity at the top y H 0 At the highest point of the trajectory,. Hint 2. Which equation to use The three kinematic equations that govern the motion in the y direction are v y v y gt v0y,

5 and y + t g y 0 v0y 1 2 t 2, v 2 y 2gy v 2 0y. The third equation contains the height variable y, and all the other quantities are known (at y H), so you could use it to find H. This would be the simplest method. Alternately, if you prefer, you could first find the time required to reach the maximum height from the first equation, and then use this time in the second equation to solve for H. H 2 ( v0sin(θ)) 2g Part B What is the range of the projectile, R? Express the range in terms of v0, θ, and g. Hint 1. Find the total time spent in air Find the total time the projectile spends in the air, by considering the time (found in ) and then the time it takes to fall back to the ground. Express your answer in terms of v0, θ, and g. t H it takes to reach the highest point t R 2 sin(θ) v0 g Hint 2. Find x(t) What is the x coordinate of the projectile's position? Express your answer in terms of t, v0, and θ. x(t) cos(θ)t v0 R 2( sin(θ)cos(θ)) v0 2 g

6 Consider your advice to an artillery officer who has the following problem. From his current postition, he must shoot over a hill of height H at a target on the other side, which has the same elevation as his gun. He knows from his accurate map both the bearing and the distance R to the target and also that the hill is halfway to the target. To shoot as accurately as possible, he wants the projectile to just barely pass above the hill. Part C Find the angle θ above the horizontal at which the projectile should be fired. Express your answer in terms of H and R. Hint 1. How to approach the problem In the first half of this problem, you found H and R in terms of v0 and θ. Solve these two equations to find θ in terms of H and R. Hint 2. Set up the ratio Find the ratio of H to R. The only variable in your answer should be θ. H/R tan(θ) 4 4H θ atan( ) R

7 Recall the following trigonometry formulas: and sinθ opposite hypotenuse cosθ adjacent hypotenuse tanθ oppposite adjacent.,, tan(θ) 4H/R In this case, since, you can draw a right triangle with θ as one of the angles, an "opposite" side of length 4H, and an "adjacent" side of length R. You can then use this triangle to find sin(θ) and cos(θ), after you find the length of the hypotenuse using the Pythagorean Theorem. Part D What is the initial speed? Express v0 in terms of g, R, and H. Hint 1. How to approach this part Use one of the equations that you had derived for and. You will need to find an expression for and/or cos(θ) to find v0. Hint 2. Find sin(θ) tan(θ) Use the expression you derived for and the Pythagorean theorem to find. Leave your answer in terms of H and R H R sin(θ) sin(θ) sin(θ) 4H R 2 +16H 2 Hint 3. Find cos(θ) cos(θ) Now find the expression for. Leave your answer in terms of H and R.

8 cos(θ) R R 2 +16H 2 v0 g R2+16 H 2 8H Part E Find tg, the flight time of the projectile. Express the flight time in terms of H and g. Hint 1. How to proceed First, find tg in terms of v0 and θ. You can use hints in Part B in the previous half. tg 2 sin(θ) v0 g tg 2 2gH g Score Summary: Your score on this assignment is 101%. You received 5.02 out of a possible total of 5 points.