#$&* course PHY 201 4:17 PM on 2/17/13 008. Using the Acceleration of Gravity; summarizing the analysis of uniformly accelerated motion
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Given Solution: You would have an initial vertical velocity of 0, and would accelerate at 9.8 m/s^2 in the same direction as your 1 meter vertical displacement. You would also have a slight horizontal velocity (you don't step off of a table without moving a bit in the horizontal direction, and you would very likely maintain a small horizontal velocity as you fell), but this would have no effect on your vertical motion. So your vertical velocity is a uniform acceleration with v0 = 0, `ds = 1 meter and a = 9.8 m/s^2. The equation vf^2 = v0^2 + 2 a `ds contains the three known variables and can therefore be used to find the desired final velocity. We obtain vf = +- `sqrt( v0^2 + 2 a `ds) = +- `sqrt ( 0^2 + 2 * 9.8 m/s^2 * 1 m)= +- `sqrt ( 19.6 m^2 / s^2) = +- 4.4 m/s, approx. Since the acceleration and displacement were in the direction chosen as positive, we conclude that the final velocity will be in the same direction and we choose the solution vf = +4.4 m/s. STUDENT QUESTION ????how important is the plus or minus before a square root or an approximation of a square root and what is its significance???? INSTRUCTOR RESPONSE: In some situations the + solution will be the one that fits the situation, while in others it will be the -, and in others both the + and - solutions can be valid. An example: If a ball is thrown into the air, rising say 20 meters before falling back, then it will be at the 15 meter height twice, once while rising and once while falling. At one of those times the velocity will be positive, at the other the velocity will be negative. So both the + and the - solutions are valid. On the other hand you might want the velocity when the ball hits the ground. Only for a downward velocity will be valid for this question. Your equations will give you the + and the - solution, and you will have to determine which applies (i.e., which is downward, given your choice of the positive direction).*@ &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `q002. If you jump vertically upward, leaving the ground with a vertical velocity of 3 m/s, how high will you be at the highest point of your jump? Note that as soon as you leave the ground, you are under the influence of only the gravitational force. All the forces that you exerted with your legs and other parts of your body to attain the 3 m/s velocity have done their work and are no longer acting on you. All you have to show for it is that 3 m/s velocity. So as soon as you leave the ground, you begin experiencing an acceleration of 9.8 m/s^2 in the downward direction. Now again, how high will you be at the highest point of your jump? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: v0 = 3 m/s, 'ds = ?, a = -9.8 m/s^2, vf = 0 m/s vf^2 = v0^2 + 2 a `ds => 'ds = (vf - v0)^2 / 2a = (0 m/s - 3 m/s)^2 / (2 * -9.8 m/s^2) = (-9 m^2/s^2) / -19.6 m/s^2 = 0.46 m confidence rating #$&*:8232; ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: From the instant the leave the ground until the instant you reach your highest point, you have an acceleration of 9.8 m/s^2 in the downward direction. Since you are jumping upward, and since we can take our choice of whether upward or downward is the positive direction, we choose the upward direction as positive. You might have chosen the downward direction, and we will see in a moment how you should have proceeded after doing so. For now, using the upward direction as positive, we see that you have an initial velocity of v0 = + 3 m/s and an acceleration of a = -9.8 m/s^2. In order to use any of the equations of motion, each of which involves four variables, you should have the values of three variables. So far you only have two, v0 and a. {}What other variable might you know? If you think about it, you will notice that when objects tossed in the air reach their highest point they stop for an instant before falling back down. That is precisely what will happen to you. At the highest point your velocity will be 0. Since the highest point is the last point we are considering, we see that for your motion from the ground to the highest point, vf = 0. Therefore we are modeling a uniform acceleration situation with v0 = +3 m/s, a = -9.8 m/s^2 and vf = 0. We wish to find the displacement `ds. Unfortunately none of the equations of uniformly accelerated motion contain the four variables v0, a, vf and `ds. This situation can be easily reasoned out from an understanding of the basic quantities. We can find the change in velocity to be -3 meters/second; since the acceleration is equal to the change in velocity divided by the time interval we quickly determine that the time interval is equal to the change in velocity divided by the acceleration, which is `dt = -3 m/s / (-9.8 m/s^2) = .3 sec, approx.; then we multiply the .3 second time interval by the 1.5 m/s average velocity to obtain `ds = .45 meters. However if we wish to use the equations, we can begin with the equation vf = v0 + a `dt and solve to find `dt = (vf - v0) / a = (0 - 3 m/s) / (-9.8 m/s^2) = .3 sec. We can then use the equation `ds = (vf + v0) / 2 * `dt = (3 m/s + 0 m/s) / 2 * .3 sec = .45 m. This solution closely parallels and is completely equivalent to the direct reasoning process, and shows that and initial velocity of 3 meters/second should carry a jumper to a vertical height of .45 meters, approximately 18 inches. This is a fairly average vertical jump. If the negative direction had been chosen as positive then we would have a = +9.8 m/s^2, v0 = -3 m/s^2 (v0 is be in the direction opposite the acceleration so if acceleration is positive then initial velocity is negative) and again vf = 0 m/s (0 m/s is the same whether going up or down). The steps of the solution will be the same and the same result will be obtained, except that `ds will be -.45 m--a negative displacement, but where the positive direction is down. That is we move .45 m in the direction opposite to positive, meaning we move .45 meters upward. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: ********************************************* Question: `q003. If you roll a ball along a horizontal table so that it rolls off the edge of the table at a velocity of 3 m/s, the ball will continue traveling in the horizontal direction without changing its velocity appreciably, and at the same time will fall to the floor in the same time as it would had it been simply dropped from the edge of the table. If the vertical distance from the edge of the table to the floor is .9 meters, then how far will the ball travel in the horizontal direction as it falls? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: v0 = 3 m/s (horizontal), v0 = 0 m/s (vertical), 'ds = 0.9 m (vertical), a = 9.8 m/s^2 (downward), 'dt = ? 'ds = v0'dt + 0.5a'dt^2 => 'ds = 0.5 a 'dt^2 'dt = sqrt('ds / 0.5a) = sqrt(0.9 m / (0.5(9.8 m/s^2)) = 0.43 s The horizontal velocity remains constant at 3 m/s, therefore: vAve = 3 m/s vAve = 'ds / 'dt => 'ds = 'dt * vAve = 0.43 s * 3 m/s = 1.29 m The ball will travel approximately 1.29 m in the horizontal direction as it falls. It will fall to the floor in the same time as it would had it been dropped from the edge of the table. confidence rating #$&*:8232; ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^
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Given Solution: A ball dropped from rest at a height of .9 meters will fall to the ground with a uniform vertical acceleration of 9.8 m/s^2 downward. Selecting the downward direction as positive we have `ds = .9 meters, a = 9.8 m/s^2 and v0 = 0. Using the equation `ds = v0 `dt + .5 a `dt^2 we see that v0 = 0 simplifies the equation to `ds = .5 a `dt^2, so `dt = `sqrt( 2 `ds / a) = `sqrt(2 * .9 m / (9.8 m/s^2) ) = .42 sec, approx.. Since the ball rolls off the edge of the table with only a horizontal velocity, its initial vertical velocity is still zero and it still falls to the floor in .42 seconds. Since its horizontal velocity remains at 3 m/s, it travels through a displacement of 3 m/s * .42 sec = 1.26 meters in this time. STUDENT COMMENT: I don’t see why you don’t use the formula that I used in my answer. INSTRUCTOR RESPONSE: In your solution you chose to solve the third equation ds = v0 * dt + 0.5 * a * dt^2 for `dt. Since v0 = 0 for the vertical motion, the equation simplifies and is easy to solve for `dt. However I avoid using the third equation to solve for `dt because it is quadratic in `dt, and is therefore very confusing to most students. It is less confusing to use the fourth equation to find vf, after which we can easily reason out `dt. Of course in the present case v0 = 0 and the equation becomes every bit as simple as the fourth equation; in fact when v0 = 0 it's simpler to use the equation you used. However most students have problems with special conditions and special cases, so I choose not to confuse the issue, and consistently use the fourth equation in my solutions. My convention of using the fourth equation in this situation does no harm to students who understand how to solve the quadratic, and who know how consider and apply special conditions. You and other students who are sufficiently comfortable with the mathematics should always use the most appropriate option. &&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&&& Self-critique (if necessary): ------------------------------------------------ Self-critique rating: You should submit the above questions, along with your answers and self-critiques. You may then begin work on the Questions, Problem and Exercises, as instructed below. ********************************************* Question: `q004. A ball starts out with a horizontal velocity of 4 m/s and vertical velocity 0 m/s. It falls freely for 2 seconds. During this 2 seconds how far does it travel in the vertical direction, and how far in the horizontal direction? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: v0 = 4 m/s (horizontal) and 0 m/s (vertical). 'dt = 2 seconds, a = 9.8 m/s^2. Find 'ds horizontal and 'ds vertical. a = 'dv / 'dt = (4 m/s - 0 m/s) / 2 s = 2 m/s^2 'ds_horizontal = v0'dt + 0.5a'dt^2 = (4 m/s)(2 s) + 0.5(2 m/s^2)(2 s)^2 = 8 m + 4 m = 12.0 m 'ds_vertical = 'ds = v0'dt + 0.5a'dt^2 => 'ds = 0.5 a 'dt^2 = 0.5(9.8 m/s^2)(2 s)^2 = 19.6 m confidence rating #$&*: ^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^^ ********************************************* Question: `q005. A ball starts out with a horizontal velocity of 4 m/s and vertical velocity 0 m/s. It falls freely to the ground 20 meters below. During its fall how far does it travel in the vertical direction, and how far in the horizontal direction? YYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYYY Your solution: v0 = 4 m/s horizontal, 0 m/s vertical, 'ds = 20 m, 'dt = ? 'ds = v0'dt + 0.5a'dt^2 => 'ds = 0.5 a 'dt^2 'dt = sqrt('ds / 0.5a) = sqrt(20 m / (0.5(9.8 m/s^2)) = 2.02 s vAve = 4 m/s horiztonal vAve = 'ds / 'dt => 'ds = vAve * 'dt = 4 m/s * 2.02 s = 8.08 m Would it not fall a total of 20 meters vertically because that is the total vertical distance to ground?