PHY 231
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A ball rolls off the end of an incline with a vertical velocity of 20 cm/s downward, and a horizontal velocity of 80 cm/s. The ball falls freely to the floor 120 cm below.
• For the interval between the end of the ramp and the floor, what are the ball's initial velocity, displacement and acceleration in the vertical direction?
The ball's initial vertical velocity is a -20 cm/s. Its initial displacement is 0 cm before it starts. When it hits the ground, its ultimate displacement is 120 cm below where it began to roll off the end of the incline. And its initial acceleration is 9.8 m/s^2, as applies to all freely falling objects.
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• What therefore are its final velocity, displacement, change in velocity and average velocity in the vertical direction?
It's final velocity is 0 m/s, when it hits the ground. It's final displacement is 120 cm below its beginning position. The change in velocity in the vertical direction is about 59 cm/s ( final ( when it hits the ground ) - initial ). And its average velocity is 50 cm/s (( 20 cm/s + 79.277652 cm/s )/2 ).
Calculations:
6.04874 s * 9.8 m/s^2 = 79.277652 m/s
79.277652 m/s - 20 m/s = 59.277652 m/s
( 20 cm/s + 79.277652 cm/s ) = 49.638826 m/s
Time to hit ground calculation below...
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• What are the ball's acceleration and initial velocity in the horizontal direction, and what is the change in clock time, during this interval?
The ball's acceleration in the horizontal direction is constant, or 0 cm/s^2, as long as air resistance is neglected. The initial velocity is 80 cm/s in horizontal direction.
Quadratic Formula for Time:
0 = 9.8t -4.9t^2 + 120
0 = -4.9t^2 + 9.8t + 120
t = -4.04874 s or 6.04874 s
It doesn't take anywhere near that long. The problem is that you didn't include units in your calculation, and there are inconsistencies in your units.
It has to be positive, taking a little over six seconds ( 6.01 s to be more exact ) for the ball to hit the ground. Thus, that is the time interval.
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• What therefore are its displacement, final velocity, average velocity and change in velocity in the horizontal direction during this interval?
It's displacement in the horizontal direction is 483.8992 ( 80 cm/s * 6.04874 s ), or about 484 meters. It's average velocity is about 80 cm/s. And its change in velocity is 0 cm/s.
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• After the instant of impact with the floor, can we expect that the ball will be uniformly accelerated?
No, notwithstanding the floor's surface condition and material, the ball's acceleration will decrease at an increasing rate after its initial contact with the surface, especially considering friction until it comes to a rest.
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• Why does this analysis stop at the instant of impact with the floor?
We are unable to account for, at least thus far, the acceleration and velocity changes after impact with the floor. We cannot accurately model and simulate the ball's motion. Thus, no further analysis may be done.
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30 Minutes
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You have an inconsistency in your units, when calculating time of fall.
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