Concepts & Calculations Group Learning Problems

Note: Each of these problems consists of Concept Questions followed by a related quantitative Problem. They are designed for use by students working alone or in small learning groups. The Concept Questions involve little or no mathematics and are intended to stimulate group discussions. They focus on the concepts with which the problems deal. Recognizing the concepts is the essential initial step in any problem-solving technique.

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Concept Questions A puck is moving on an air hockey table. Relative to an x, y coordinate system at time t=0 s, the x components of the puck’s initial velocity and acceleration are v_0x=+1.0 m/s and a_x=+2.0 m/s². The y components of the initial velocity and acceleration are v_0y=+12.0 m/s and a_y=–2.0 m/s². (a) Is the magnitude of the x component of the velocity increasing or decreasing in time? (b) Is the magnitude of the y component of the velocity increasing or decreasing in time?

Problem Find the magnitude and direction of the puck’s velocity at a time of t=0.50 s. Specify the direction relative to the +x axis. Be sure that your calculations are consistent with your answers to the Concept Questions.

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Concept Questions (a) A projectile is launched at a speed v₀ and at an angle above the horizontal; its initial velocity components are v_0x and v_0y. In the absence of air resistance, what is the speed of the projectile at the peak of its trajectory? (b) What is its speed just before it lands at the same vertical level from which it was launched? (c) Consider its speed at a point that is at a vertical level between that in (a) and (b). How does the speed at this point compare with the speeds identified in (a) and (b)? In each case, give your reasoning.

Problem A golfer hits a shot to a green that is elevated 3.0 m above the point where the ball is struck. The ball leaves the club at a speed of 14.0 m/s at an angle of 40.0° above the horizontal. It rises to its maximum height and then falls down to the green. Ignoring air resistance, find the speed of the ball just before it lands. Check to see that your answer is consistent with your answer to part (c) of the Concept Questions.

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Concept Questions (a) When a projectile is launched horizontally from a rooftop at a speed v_0x, does its horizontal velocity component ever change in the absence of air resistance? (b) Can you calculate the horizontal distance D traveled after launch simply as D=v_0xt, where t is the fall time of the projectile? (c) In calculating the fall time, is the vertical part of the motion just like that of a ball dropped from rest? Explain each of your answers.

Problem A criminal is escaping across a rooftop and runs off the roof horizontally at a speed of 5.3 m/s, hoping to land on the roof of an adjacent building. The horizontal distance between the two buildings is D, and the roof of the adjacent building is 2.0 m below the jumping-off point. Find the maximum value for D.

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Concept Questions (a) When a projectile is thrown at a speed v₀ at an angle above the horizontal, which component of the launch velocity determines the maximum height reached by the projectile? (b) Do the horizontal and vertical components of the launch velocity have larger or smaller magnitudes than v₀? (c) If the projectile is thrown straight upward at a speed v₀, does it attain a greater or lesser height than that in part (a)? In each case, account for your answer.

Problem A ball is thrown upward at a speed v₀ at an angle of 52° above the horizontal. It reaches a maximum height of 7.5 m. How high would this ball go if it were thrown straight upward at speed v₀? Verify that your answer is consistent with your answer to part (c) of the Concept Questions.

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Concept Questions (a) For a projectile that follows a trajectory like that in Figure 3.12 the range is given by R=v_0xt, where v_0x is the horizontal component of the launch velocity and t is the time of flight. Is v_0x proportional to the launch speed v₀? (b) Is the time of flight t proportional to the launch speed v₀? (c) Is the range R proportional to v₀ or v₀²? Explain each answer.

Problem In the absence of air resistance, a projectile is launched from and returns to ground level. It follows a trajectory similar to that in Figure 3.12 and has a range of 23 m. Suppose the launch speed is doubled, and the projectile is fired at the same angle above the ground. What is the new range?

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Concept Questions Two friends, Barbara and Neil, are out rollerblading. With respect to the ground, Barbara is skating due south. Neil is in front of her and to her left. With respect to the ground, he is skating due west. (a) Does Barbara see him moving toward the east or toward the west? (b) Does Barbara see him moving toward the north or toward the south? (c) Considering your answers to parts (a) and (b), how does Barbara see Neil moving relative to herself, toward the east and north, toward the east and south, toward the west and north, or toward the west and south? Justify your answers in each case.

Problem With respect to the ground, Barbara is skating due south at a speed of 4.0 m/s. With respect to the ground, Neil is skating due west at a speed of 3.2 m/s. Find Neil’s velocity (magnitude and direction relative to due west) as seen by Barbara. Make sure that your answer agrees with your answer to part (c) of the Concept Questions.