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Velocity describes how fast an object changes its position. The higher the velocity the quicker the object travels between locations.
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Acceleration tells how fast an object changes its velocity. Phrases like slow down, speed up, change speed, and change velocity are used to describe accelerations. If someone wants an easy way to determine if he or she visualizing acceleration or constant velocity a long straight line he needs to ask himself one question, " Is the object going slower or faster."
Roller coasters trade height for velocity and velocity for height. Most calculations rely on using velocity measurements in one way or another. The first step is being able to calculate changes in speed. In the ideal world energy is conserved. Forces are ignored in early designed stages. Mechanical energy on a coaster comes in two forms.
Kinetic energy, and KE=1/2 mv2, and potential energy, PE = mgh.
Total energy, is conserved and is equal to the sum of Kinetic and potential at any single location.
ET=KE+PE (at any single location)
Example 1
What is the velocity at the bottom of the second hill?
Solution:
ET (TOP) = ET (BOTTOM)
KE + PE =KE + PE
(1/2) mv2 + MGH= (½)
The masses cancel out because it is the same coaster at the top and bottom.
(½) v2 + GH = (½) v2 + GH
Substitute the numbers at each location.
(1/2) (8.8) 2 + 9.8 (95) = (½) v2 + 9.8 (0)
The height at the bottom is zero because it is the lowest point when comparing to the starting height.
38.72 + 931 = (½) v2
969.72= (½) v2
1939.44= v2
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v = 44.04 m/s … at the bottom of the first hill. |
Example 2
What is the velocity at the top of the second hill?
Solution:
ET ( top of first hill ) = ET ( top of second hill )
KE + PE = KE + PE
( 1/2 ) mv2+ mgh = ( 1/2 ) mv2 + mgh
(1/2) v2 + gh = (1/2) mv2+ mgh
The masses cancel out because it is the same coaster at the top and bottom.
(1/2) v2 + gh = (1/2) v2 + gh
Substitute the number at each location.
(1/2) (8.8) 2 + 9.8 (95) = (1/2) v2 + 9.8 (65)
Notice all the numbers on the left side come from the top of the first hill while all the numbers on the side come from the top of the second hill.
38.72 + 931 = (1/2) v2 + 637
332.72 = (1/2) v2
665.44 = v2
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v = 25.80 m/s …at the top of the second hill. |
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