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Seed quest 8.1
A ball is tossed upward with an initial velocity of 25 meters / second. Assume that the acceleration of gravity is 10 m/s^2 downward.
What will be the velocity of the ball after one second?
answer/question/discussion:
v0 = 25 m/s
a = -10 m/s/s
dt = 1s
vf = v0 +a*dt
vf = 25m/s - 10m/s/s *1s
vf = 25m/s - 10m/s = 15 m/s
I believe that the acceleration is negative because the problem states that the acceleration is downward, but the ball was thrown upward.
Right.
You don't really need the equations. In 1 second the ball's velocity decreases by 10 m/s.
Be sure you understand this both ways.
What will be its velocity at the end of two seconds?
answer/question/discussion:
v0 = 25 m/s
a = -10m/s/s
dt = 2s
vf = v0 + a*dt
vf = 25m/s - 10 m/s/s * 2s
vf = 25m/s - 20 m/s = 5 m/s
During the first two seconds, what therefore is its average velocity?
answer/question/discussion:
v0 = 25m/s
vf = 5 m/s
vAve = (v0 + vf)/2
vAve = (25m/s + 5m/s) /2 = 30m/s / 2 = 15 m/s
How far does it therefore rise in the first two seconds?
answer/question/discussion:
vAve = 15m/s
dt = 2 s
ds = vAve * dt
ds = 15m/s * 2s = 30m
What will be its velocity at the end of a additional second, and at the end of one more additional second?
answer/question/discussion:
Vf = 25m/s - 10m/s/s * 3 s
Vf = 25m/s - 30 m/s = -5m/s
Vf = 25m/s - 10m/s/s * 4s
Vf = 25m/s - 40 m/s = -15m/s
These numbers indicate that the ball is now falling back down.
At what instant does the ball reach its maximum height, and how high has it risen by that instant?
answer/question/discussion:
At maximum height the vf would = 0
V0 = 25m/s
A = -10m/s/s
Vf = 0
Solve for dt
Vf = v0 +a* dt
0m/s = 25m/s - 10m/s/s * dt
??? How do I do this???
You could rearrange to get 10 m/s^2 * `dt = 25 m/s, so that `dt = 25 m/s / (10 m/s^2) = 2.5 s.
Probably better to solve the equation symbolically:
vf = v0 + a `dt so
a `dt = vf - v0 so
`dt = (vf - v0) / a = 25 m/s / (10 m/s^2) = 2.5 s.
I will use logic instead…
At the end of 2 sec the v = 5 m/s
At the end of 3 sec the v = -5 m/s
Therefore it stands to reason that halfway between the two at 2.5 sec the v = 0. To check this answer I will plug in 2.5 s into the equation and see if I get vf = 0 Good thinking.
Vf = v0 + a*dt
Vf = 25m/s - 10 m/s/s * 2.5s
Vf = 25m/s - 25m/s = 0
The ball reaches maximum height at 2.5secs.
What is the maximum height?
V0 = 25m/s
Vf = 0m/s
Dt = 2.5s
Ds = (v0 + vf)/2 * dt
Ds = (25m/s + 0m/s) / 2 * 2.5s
Ds = 25m/s / 2 * 2.5s
Ds = 12.5m/s * 2.5s = 31.25m
The ball will have reached a height of 31.25m at the maximum height in 2.5 secs.
What is its average velocity for the first four seconds, and how high is it at the end of the fourth second?
answer/question/discussion:
Vave = (v0 + vf) /2
Vave = (25m/s - 15 m/s) /2
Vave = 5m/s /2 = 2.5m/s
Ds = vAve * dt
Ds = 2.5m/s * 4 sec = 10 m
How high will it be at the end of the sixth second?
answer/question/discussion:
V0 = 25m/s
A = -10m/s/s
Dt = 6 s
Vf = 25m/s - 10m/s/s * 6 s
Vf = 25m/s - 60m/s = -35m/s
Ds = (v0 + vf)/2 * dt
Ds = (25m/s - 35m/s) /2 * 6 s
Ds = -10m/s / 2 * 6s
Ds = -5m/s * 6 = -30m/s
The maximum height was 31.25. Subtract 30. The height at the end of 6 sec = 1.25m
The six-second interval did not start at the maximum height. It started when the velocity was +25 m/s, which is what you used to find the average velocity.
The -30 m displacement is relative to the initial height.
Now, I don’t think this is right. It seems that I should start over at the maximum height and the acceleration should now be positive10m/s/s because the ball is now traveling downward.
V0 would = 0m/s ( at the maximum height it stops)
A = 10m/s/s
Dt = 6 s - 2.5s ( it took 2.5 to get to the maximum and now we are starting over) = 3.5s
Vf = 0m/s + 10m/s/s * 3.5s
Vf = 10 m/s/s * 3.5s
Vf = 35m/s
Ds = (v0 + vf)/2 * dt
Ds = (0m/s + 35m/s) /2 * 3.5s
Ds = 35m/s / 2 * 3.5s
Ds = 17.5m/s * 3.5s = 61.25
Well, that’s not possible because the ball only went up 31.25 m, it can’t go down 61.25m…I think I will stick with my first answer! The height at the end of 6 s = 1.25m, but I have very little confidence in it.
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30 mins?
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Please respond with clarification soon. I don't want to attempt seed question 8.2 without feedback from this one.
Very good work. However, be sure to see my notes, especially related to the last question.