The paths they take may be spiral-like. You can model the spiral motion as uniform circular motion combined with a constant upward velocity. Assume a bird completes a circle of radius 8. 00 m every 5. 00 s and rises vertically at a rate of 3. 00 m/s. Find the speed of the bird relative to the ground. 10. 5 Find the magnitude of the bird’s acceleration. 12. 6 Part C Find the direction of the bird’s acceleration. 0. 0 above the horizontal Part D Find the angle between the bird’s velocity vector and the horizontal. ANSWER: 16. 6 Problem 3. 9 Two tanks are engaged in a training exercise on level ground.

The first tank fires a paint-filled training round with a muzzle speed of 239 at an angle 10. 8 above the horizontal while advancing toward the second tank with a speed of 16. 5 speed of 34. 0 relative to the ground. The second tank is retreating at a relative to the ground, but is hit by the shell. You can ignore air resistance and assume the shell hits at the same height above ground from which it was fired. Find the distance between the tanks when the round was first fired. Take free fall acceleration to be = 9. 80 1990 Find the distance between the tanks at the time of impact. 150 Battleship Shells A battleship simultaneously fires two shells toward two identical enemy ships. One shell hits ship A, which is close by, and the other hits ship B, which is farther away. The two shells are fired at the same speed. Assume that air resistance is negligible and that the magnitude of the acceleration due to gravity is . What shape is the trajectory (graph of y vs.. X ) of the shells? ANSWER: straight line parabola hyperbola The shape cannot be determined. Part B For two shells fired at the same speed which statement about the horizontal stance traveled is correct?

Hint 8. 1 Two things to consider The shell fired at a larger angle with respect to the horizontal lands farther away. The shell fired at an angle closest to 45 degrees lands farther away. The shell fired at a smaller angle with respect to the horizontal lands farther away. The lighter shell lands farther away. Consider the situation in which both shells are fired at an angle greater than 45 degrees with respect to the horizontal. Remember that enemy ship A is closer than enemy ship B. Part C Which shell is fired at the larger angle? Hint c. i Consider the limiting case Hint C. I

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Both shells are fired at the same angle. Which shell is launched with a greater vertical velocity, Both shells are launched with the same vertical velocity. Part E Which shell is launched with a greater horizontal velocity, Both shells are launched with the same horizontal velocity. Part F Which shell reaches the greater maximum height? Hint RI What determines maximum height? Both shells reach the same maximum height. Part G Which shell has the longest travel time (time elapsed between being fired and hitting the enemy ship)? Hint G. 1 Consider the limiting case Both shells have the same travel time. A Wild Ride

A car in a roller coaster moves along a track that consists of a sequence of ups and downs. Let the x axis be parallel to the ground and the positive y axis point upward. In the time interval from to s, the trajectory of the car along a certain section of the track is given by , where is a positive dimensionless constant. At is the roller coaster car ascending or descending? How to approach the problem Find the vertical component of the velocity of the car ascending descending Derive a general expression for the speed of the car. Hint B. 2 Magnitude of a vector Hint B. 3 Find the components of the velocity of the car

Hint 8. 3 Express your answer in meters per second in terms of and . The roller coaster is designed according to safety regulations that prohibit the speed of the car from exceeding . Find the maximum value of allowed by these regulations. To comply with the regulations, the speed of the car cannot exceed the given safety limit at any time. Thus, you need to determine what the maximum value of the speed is and impose the condition that such a value cannot be greater than the safety limit. Hint C. 2 Find the maximum value of the speed Given the expression found in Part B, find the maximum speed f the car in terms of Hint C. . 1 Using the calculus Hint C. 2. 2 Find the first derivative of the speed Hint C. 2. 3 Find the time at which the speed reaches its maximum value Express your answer in meters per second. = Answer not displayed Express your answer using two significant figures. = 1. 7 Shooting over a Hill A projectile is fired with speed at an angle from the horizontal as shown in the figure . Part A Find the highest point in the trajectory, Hint A. 1 Velocity at the top Which equation to use Express the highest point in terms of the magnitude of the acceleration due to ravine , the initial velocity , and the angle .

What is the range of the projectile, Find the total time spent in air Express the range in terms of , and Consider your advice to an artillery officer who has the following problem. From his current position, he his current position, he must shoot over a hill of height 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 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.

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