Think about Galileo\'sinclined plane experiments. a) If a metal marble rolls 15

Question

Think about Galileo'sinclined plane experiments.
a) If a metal marble rolls 15 cm in the first unit of time, how farwill it roll in the third unit of time?
b) If the units of time were seconds, what is the speed of the ballafter the first unit of time? After the third unit?
What is the acceleration?
c) We did not derive Galileo's equations of motion. However, we maywant to verify that they are at least possible.
One quick check is that they should be dimensionally accurate.Confirm with the equations below: Vf=Vo+at d=(Vf2-Vo2)/2a d=Vot+1/2at2 Think about Galileo'sinclined plane experiments.
a) If a metal marble rolls 15 cm in the first unit of time, how farwill it roll in the third unit of time?
b) If the units of time were seconds, what is the speed of the ballafter the first unit of time? After the third unit?
What is the acceleration?
c) We did not derive Galileo's equations of motion. However, we maywant to verify that they are at least possible.
One quick check is that they should be dimensionally accurate.Confirm with the equations below: Vf=Vo+at d=(Vf2-Vo2)/2a d=Vot+1/2at2 Vf=Vo+at d=(Vf2-Vo2)/2a d=Vot+1/2at2

Explanation / Answer

a) Use the equation x = vot +at2/2. If we measure all of the distances fromwhen the marble is first released, so vo = 0, we havethat the distance traveled in the first unit of time is equal tox = at2/2, the distance traveled in the both thefirst and second units of time is equal to x =a(2t)2/2 = 4at2/2, and the distance traveledin the first, second, and third units of time combined is equal tox = a(3t)2/2 = 9at2/2. So, thedistance traveled in the third time is equal to 9at2/2 -4at2/2 = 5at2/2. This is five times theamount traveled in the first second, which is at2/2 =15cm, so it will move 75cm in the third unit of time. b) The average speed in the first second is equal to v = d/t =0.15m / 1s = 0.15 m/s. Since it started at rest andaccelerated uniformly, we can then say that (vf +vo)/2 = 0.15 m/s, so vf = 0.15 m/s * 2 = 0.30m/s. Since it accelerates uniformly, the speed at the end ofthe third second is going to be equal to the speed at the end ofthe first second multiplied by 3, so it will be at a speed of 0.90m/s. In one second, the speed increased from 0 m/s to 0.30m/s, so the acceleration is equal to a = (vf -vo)/t = (0.30 m/s - 0)/1s = 0.30 m/s2. c) We can check these by making sure they are dimensionallyaccurate. (m/s) = (m/s) + (m/s2)(s) = (m/s) + (m/s) =(m/s) (m) = ((m/s)2 -(m/s)2)/(m/s2) = (m2/s2- m2/s2)/(m/s2) = (m) - (m) =(m) (m) = (m/s)(s) + (m/s2)(s)2 = (m) + (m)= (m)

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