4.6 Free Fall: How Far. think! 4.7 Graphs of Motion. think! At the end of time t, the object starting from rest falls a distance d.
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1 At the end of time t, the object starting from rest falls a distance d. We used freely falling objects to describe the relationship between distance traveled, acceleration, and velocity acquired. The same principles apply to any accelerating object. Whenever an object s initial speed is zero and the acceleration a is constant, velocity and distance traveled are: think! An apple drops from a tree and hits the ground in one second. What is its speed upon striking the ground? What is its average speed during the one second? How high above ground was the apple when it first dropped? think! An apple drops from a tree and hits the ground in one second. What is its speed upon striking the ground? What is its average speed during the one second? How high above ground was the apple when it first dropped? Equations and tables are not the only way to describe relationships such as velocity and acceleration. Graphs can visually describe relationships. Answer: The speed when it strikes the ground is 10 m/s. The average speed was 5 m/s and the apple dropped from a height of 5 meters. 1
2 Speed-Versus-Time On a speed-versus-time graph, the speed v of a freely falling object can be plotted on the vertical axis and time t on the horizontal axis. For every increase of 1 s, there is the same 10 m/s increase in speed. Since the object is dropped from rest, the line starts at the origin, where both v and t are zero. If we double t, we double v; if we triple t, we triple v; and so on. The curve is a straight line, so its slope is constant. For 10 m/s of vertical change there is a horizontal change of 1 s. The slope is 10 m/s divided by 1 s, or 10 m/s 2, which is the acceleration due to gravity. The straight line shows the acceleration is constant and the value of the slope is the magnitude of the acceleration. The area under the graph is the distance the object has traveled. For a linear graph, the area is a triangle. For a nonlinear graph, you find the area by performing an integral. What is the distance traveled in the first 3 seconds of the free falling object graphed at the right? Distance-Versus-Time When the distance d traveled by a freely falling object is plotted on the vertical axis and time t on the horizontal axis, the result is a curved line. The relationship between distance and time is nonlinear. The relationship is quadratic and the curve is parabolic when we double t, we do not double d; we quadruple it. Distance depends on time squared! A curved line also has a slope, called a derivative different at different points. The slope of the curve on a distance-versus-time graph is speed, the rate at which distance is covered per unit of time. The slope steepens (becomes greater) as time passes, which shows that speed increases as time passes. 2
3 4.8 Air Resistance and Falling Objects Drop a feather and a coin and the coin reaches the floor far ahead of the feather. Air resistance is responsible for these different accelerations. In a vacuum, the feather and coin fall side by side with the same acceleration, g. In many cases the effect of air resistance is small enough to be neglected. With negligible air resistance, falling objects can be considered to be falling freely. 1. Jake walks east through a passenger car on a train that moves 10 m/s in the same direction. Jake s speed relative to the car is 2 m/s. Jake s speed relative to an observer at rest outside the train is a. 2 m/s. b. 5 m/s. c. 8 m/s. d. 12 m/s. 1. Jake walks east through a passenger car on a train that moves 10 m/s in the same direction. Jake s speed relative to the car is 2 m/s. Jake s speed relative to an observer at rest outside the train is a. 2 m/s. b. 5 m/s. c. 8 m/s. d. 12 m/s. 2. A gazelle travels 2 km in a half hour. The gazelle s average speed is a. 1/2 km/h. b. 1 km/h. c. 2 km/h. d. 4 km/h. 2. A gazelle travels 2 km in a half hour. The gazelle s average speed is a. 1/2 km/h. b. 1 km/h. c. 2 km/h. d. 4 km/h. 3. Constant speed in a constant direction is a. constant velocity. b. constant acceleration. c. instantaneous speed. d. average velocity. 3
4 3. Constant speed in a constant direction is a. constant velocity. b. constant acceleration. c. instantaneous speed. d. average velocity. 4. A vehicle undergoes acceleration when it a. gains speed. b. decreases speed. c. changes direction. d. all of the above Answer: A 4. A vehicle undergoes acceleration when it a. gains speed. b. decreases speed. c. changes direction. d. all of the above 5. If a falling object gains 10 m/s each second it falls, its acceleration can be expressed as a. 10 m/s/s. b. 10 m/s 2. c. v = gt. d. both A and B. 5. If a falling object gains 10 m/s each second it falls, its acceleration can be expressed as a. 10 m/s/s. b. 10 m/s 2. c. v = gt. d. both A and B. 6. A rock falls 180 m from a cliff into the ocean. How long is it in free fall? a. 6 s b. 10 s c. 18 s d. 180 s 4
5 6. A rock falls 180 m from a cliff into the ocean. How long is it in free fall? a. 6 s b. 10 s c. 18 s d. 180 s 7. The slope of a speed-versus-time graph represents a. distance traveled. b. velocity. c. acceleration. d. air resistance. Answer: A 7. The slope of a speed-versus-time graph represents a. distance traveled. b. velocity. c. acceleration. d. air resistance. Answer: C 8. In a vacuum tube, a feather is seen to fall as fast as a coin. This is because a. gravity doesn t act in a vacuum. b. air resistance doesn t act in a vacuum. c. greater air resistance acts on the coin. d. gravity is greater in a vacuum. 8. In a vacuum tube, a feather is seen to fall as fast as a coin. This is because a. gravity doesn t act in a vacuum. b. air resistance doesn t act in a vacuum. c. greater air resistance acts on the coin. d. gravity is greater in a vacuum. 9. Speed and acceleration are actually a. one and the same concept, but expressed differently. b. rates of one another. c. entirely different concepts. d. expressions of distance traveled. Answer: B 5
6 9. Speed and acceleration are actually a. one and the same concept, but expressed differently. b. rates of one another. c. entirely different concepts. d. expressions of distance traveled. Answer: C 6
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