Give an example of a situation in which you are given v0, a and t, and reason out all possible conclusions that could be drawn from these three quantities, assuming uniform acceleration. Accompany your explanation with graphs and flow diagrams. Show how to generalize your result to obtain the symbolic expressions for s and vf.
If we are given v0, a and dt then we can find acceleration. The problem could give you only initial velocity and you would still be able to find final velocity. We could use the final velocity equation to get acceleration which is vf=v0+a*t. If we have the initial and final velocity then we can find average velocity by adding the initial velocity and final velocity divided by 2.
Most of your reasoning is good. However given v0, a and `dt you know the acceleration, which is a.
Everything else you say is correct.
Then cnce we find the average velocity, we can multiply that by `dt to get the displacement.
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Problem Number 2
A straight ramp is inclined at three different slopes. The differences in elevation between one end and the other, for the different slopes, are 2.1, 4.2 and 5.8 cm.
• The time required for a cart to coast 54 cm down the ramp, starting from rest, is 1.682856 seconds on the first incline, 1.667749 seconds on the next, and 1.688278 seconds on the last.
How well do these data confirm our suspicion that the acceleration on the ramp is linearly dependent on the slope?
.4.2-2.1=2.1 5.8-4.2=1.6 This data confirms our suspicion that the acceleration on the ramp is linearly dependent on the slope, because depending on how fast the cart rolled down the ramp depended on the slope.
For each ramp you are given `dt and `ds (54 cm in every case), and you know that v0 is zero since every trial starts from rest.
So you can find the acceleration for each trial.
You are also given the 'rise' of each ramp, so you can find its slope (rise / run).
Having found accelerations and slopes, you make a graph of acceleration vs. slope and see whether it is plausible that a reasonably good straight line can be fit to the data.
Problem Number 3
A ball reaches a ramp of length 50 cm with an unknown initial velocity and accelerates uniformly along the ramp, reaching the other end in 3.8 seconds. Its velocity at the end of the ramp is determined to be 6.31579 cm/s. What is its acceleration on the ramp?
.vf=6.31579 6.31579/3.8= 1.66 cm/sec
This would be correct if the initial velocity was zero, but that is not the case here.
Find the average velocity. Then use this with the given final velocity to find the initial velocity.
Then you will be able to find the change in velocity, and use this result with the given time interval to find the acceleration.
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Problem Number 4
A projectile leaves the edge of a table and, while traveling horizontally at a constant velocity, falls freely a distance of 74 cm to the floor. It travels a horizontal distance of 5.7 cm during its fall. If its vertical acceleration is 980 cm/s2, how long does it take to fall and what is its horizontal velocity during the fall?
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5.7 cm*9.8= 55.86 cm/sec 74 cm/55.86= 1.32 m/s^2. I’m not sure how to find the horizontal velocity.
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Analyze vertical and horizontal motions separately.
Initial vertical velocity is zero, vertical accel is 980 cm/s^2 downward, vertical displacement is 74 cm.
I suggest finding final velocity use the fourth equation of motion, then reasoning out the time required to fall.
`dt is the same for vertical and horizontal motions
horizontal velocity is unchanging, so the average horizontal velocity is equal to the initial and final horizontal velocities
you know the horizontal displacement and now know the time interval `dt, so you can easily find vAve for the horizontal motion
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Problem Number 5
An object is given an unknown initial velocity up a ramp on which its acceleration is known to have magnitude 12 cm/s^2. It reaches a maximum distance of 228.16 cm up the ramp before rolling back down.
• What is its initial velocity?
??In order to find this would I first do 228.16 cm/12 cm/s^2=12.01 cm/sec?? Would this then be the initial velocity?
This won't work; `ds / a is not a useful quantity (note also that the units don't work out correctly).
The key is to identify the quantities you are give. You are given `ds and a. You still need a third quantity.
You are told that the object is at maximum displacement, which occurs then it stops for an instant before beginning to roll back down. Thus vf = 0.
So you know vf, a and `ds. You can solve the problem using the equations of motion and/or definitions of velocity and acceleration.
• How many seconds after the initial impetus does the object pass a point 7.9 cm up the ramp from its initial position (give all possible answers)? I’m confused about this problem.
Having found v0 for this situation, you are given `ds = 7.9 cm. You're still on the same ramp so the acceleration is the same as before.
Thus you know v0, a and `ds. Again, use the equations and/or definitions to find vf and `dt.