Just so I can narrow down the information I study, which assignments does Test 1 cover?
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18:52:40 `q001. Note that this assignment contains 3 questions. . A projectile has an initial velocity of 12 meters/second, entirely in the horizontal direction. After falling to a level floor three meters lower than the initial position, what will be the magnitude and direction of the projectile's velocity vector?
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RESPONSE --> Horizontal vo = 12m/s `ds = 3m vf^2 = vo^2 + 2a`ds Initial velocity = 0m/s vf^2 = 2(9.8m/s^2)(3m) vf^2 = 58.8m^2/s^2 vf = +-7.67 m/s, The final velocity will be negative since it is in the downward direction on the negative y-axis. We can picture the initial and final velocities as the legs of a right triangle. Therefore, the hypotenuse represents the velocity vector. To find the magnitude of that vector, we use the Pythagorean Theorem. c^2 = (12m/s)^2 + (-7.7m/s)^2 = 203.29m^2/s^2 c = 14.2m/s To find the direction of the velocity, we use the equation: theta = tan-1(y component/x component) + 360. We add 360 degrees because the y component is negative. theta = tan-1(-7.7m/s / 12m/s) + 360 = 327.3 degrees
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18:53:12 To answer this question we must first determine the horizontal and vertical velocities of the projectile at the instant it first encounters the floor. The horizontal velocity will remain at 12 meters/second. The vertical velocity will be the velocity attained by a falling object which is released from rest and allowed to fall three meters under the influence of gravity. Thus the vertical motion will be characterized by initial velocity v0 = 0, displacement `ds = 3 meters and acceleration a = 9.8 meters/second ^ 2. The fourth equation of motion, vf^2 = v0^2 + 2 a `ds, yields final vel in y direction: vf = +-`sqrt( 0^2 + 2 * 9.8 meters/second ^ 2 * 3 meters) = +-7.7 meters/second. Since we took the acceleration to be in the positive direction the final velocity will be + 7.7 meters/second. This final velocity is in the downward direction. On a standard x-y coordinate system, this velocity will be directed along the negative y axis and the final velocity will have y coordinate -7.7 m/s and x coordinate 12 meters/second. The magnitude of the final velocity is therefore `sqrt((12 meters/second) ^ 2 + (-7.7 meters/second) ^ 2 ) = 14.2 meters/second, approximately. The direction of the final velocity will therefore be arctan ( (-7.7 meters/second) / (12 meters/second) ) = -35 degrees, very approximately, as measured in the counterclockwise direction from the positive x axis. The direction of the projectile at this instant is therefore 35 degrees below horizontal. This angle is more commonly expressed as 360 degrees + (-35 degrees) = 325 degrees.
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RESPONSE --> I understand.
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18:56:31 `q002. A projectile is given an initial velocity of 20 meters/second at an angle of 30 degrees above horizontal, at an altitude of 12 meters above a level surface. How long does it take for the projectile to reach the level surface?
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RESPONSE --> vo = 20m/s theta = 30 degrees `ds = -12 m. The displacement and acceleration are negative because they are in the downward direction. a = -9.8m/s^2 Initial velocity y component = (20m/s)(sin 30) = 10m/s vf^2 = (10m/s)^2 + 2(-9.8m/s^2)(-12m) = 335.2m^2/s^2 vf = -18.3 m/s. The final velocity is negative because it is in the downward direction. `dt = `ds/vAve We know the displacement. Now to find the average velocity we add -18.3m/s and 10m/s, then divide by 2. The average velocity is -4.3m/s. `dt = (-12m)/(-4.3m/s) = 2.79 seconds.
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18:57:39 To determine the time required to reach the level surface we need only analyze the vertical motion of the projectile. The acceleration in the vertical direction will be 9.8 meters/second ^ 2 in the downward direction, and the displacement will be 12 meters in the downward direction. Taking the initial velocity to be upward into the right, we situate our x-y coordinate system with the y direction vertically upward and the x direction toward the right. Thus the initial velocity in the vertical direction will be equal to the y component of the initial velocity, which is v0y = 20 meters/second * sine (30 degrees) = 10 meters/second. Characterizing the vertical motion by v0 = 10 meters/second, `ds = -12 meters (`ds is downward while the initial velocity is upward, so a positive initial velocity implies a negative displacement), and a = -9.8 meters/second ^ 2, we see that we can find the time `dt required to reach the level surface using either the third equation of motion `ds = v0 `dt + .5 a `dt^2, or we can use the fourth equation vf^2 = v0^2 + 2 a `ds to find vf after which we can easily find `dt. To avoid having to solve a quadratic in `dt we choose to start with the fourth equation. We obtain vf = +-`sqrt ( (10 meters/second) ^ 2 + 2 * (-9.8 meters/second ^ 2) * (-12 meters) ) = +-18.3 meters/second, approximately. Since we know that the final velocity will be in the downward direction, we choose vf = -18.3 meters/second. We can now find the average velocity in the y direction. Averaging the initial 10 meters/second with the final -18.3 meters/second, we see that the average vertical velocity is -4.2 meters/second. Thus the time required for the -12 meters displacement is `dt = `ds / vAve = -12 meters/(-4.2 meters/second) = 2.7 seconds.
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RESPONSE --> Ok.
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19:00:49 `q003. What will be the horizontal distance traveled by the projectile in the preceding exercise, from the initial instant to the instant the projectile strikes the flat surface.
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RESPONSE --> theta = 30 degrees L = 20m/s `dt = 2.7s Horizontal `ds = vAve/`dt The average horizontal velocity is the x component. So the equation we use is, x component = L*cos(theta) VAve = 20m/s * cos(30) = 17.3 m/s Now we can find the horizontal displacement: `ds = (17.3m/s)/(2.7s) = 46.7 meters
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19:02:14 The horizontal velocity of the projectile will not change so if we can find this horizontal velocity, knowing that the projectile travels for 2.7 seconds we can easily find the horizontal range. The horizontal velocity of the projectile is simply the x component of the velocity: horizontal velocity = 20 meters/second * cosine (30 degrees) = 17.3 meters/second. Moving at this rate for 2.7 seconds the projectile travels distance 17.3 meters/second * 2.7 seconds = 46 meters, approximately.
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RESPONSE --> I understand this problem.
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ŠŒjîñ¼£ÛŽæº{¸ÙƵïÔݪ܇ڨœò£ÖC£ assignment #021 021. `query 21 Physics I 07-04-2006
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23:40:24 Explain how to obtain the final speed and direction of motion of a projectile which starts with known velocity in the horizontal direction and falls a known vertical distance, using the analysis of vertical and horizontal motion and vectors.
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RESPONSE --> To obtain the final speed of a projectile, we can use the vertical motion to find the time duration and then use the time to find the final horizontal speed. First we say that the initial velocity in the vertical direction is 0. We then use the formula vf^2 = 2 a * `ds. Then we use the velocity and acceleration to find the time duration.
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23:40:59 ** The horizontal velocity is unchanging so the horizontal component is always equal to the known initial horizontal velocity. The vertical velocity starts at 0, with acceleration thru a known distance at 9.8 m/s^2 downward. The final vertical velocity is easily found using the fourth equation of motion. We therefore know the x (horizontal) and y (vertical) components of the velocity. Using the Pythagorean Theorem and arctan (vy / vx) we find the speed and direction of the motion. **
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RESPONSE --> I understand this problem.
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23:43:01 Give at least three examples of vector quantities for which we might wish to find the components from magnitude and direction. Explain the meaning of the magnitude and the direction of each, and explain the meaning of the vector components.
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RESPONSE --> 1. Two football players collide. One is lighter and fast, the other is heavier and slow. The magnitude of this collision tells us the velocity of the players at collision. The direction tells us the movement as a result of the impact. The vector components are the horizontal and vertical velocities that give us the resultant vector.
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23:43:35 ** GOOD STUDENT RESPONSE: Examples might include: A force acting on an object causing it to move in an angular direction. A ball falling to the ground with a certain velocity and angle. A two car collision; velocity and momentum are both vector quantities and both important for analyzing the collision.. The magnitude and directiohn of the relsultant is the velocity and direction of travel. The vector components are the horizontal and vertical components that would produce the same effect as the resultant.
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RESPONSE --> Could you tell me if I misinterpreted anything in these explanations? I don't feel very confident in my explanation of the vector components. Am I right in thinking that the magnitude is usually the velocity or force and the direction is the direction of travel?
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