class 050923 a
Text Homework:
Phy 121: Do All Level I problems in Chapter 2.
Phy 201: Do the following problems in Chapter 2: 6, 9, 11, 16, 22, 27, 37, 43, 52, 55, 58, 63
note access links to questions:
18-34-375 (Principles of Physics, General College Physics, University Physics)
11-09-453 (General College Physics, University Physics)
23-24-380 (University Physics only)
A ball projected horizontally over the edge of a table falls to another table 10 cm below while traveling 20 cm in the horizontal direction.
To analyze projectile motion we analyze vertical and horizontal motion separately. They share nothing in common but the time interval `dt. Vertical acceleration is that of gravity, horizontal acceleration in the ideal case is 0.
The same ball, projected in the identical manner, falls to the floor 120 cm below while traveling 60 cm in the horizontal direction.
What was the average horizontal velocity of each ball?
In the first case we have for vertical velocity:
Choose downward as positive.
v0 = 0 since initial direction was horizontal
a = 980 cm/s^2
`ds = 10 cm (this is the vertical displacement).
We can now find vf and `dt:
We use the equation vf^2 = v0^2 + 2 a `ds to get vf = 140 cm/s, approx.
We find that vAve = (vf + v0) / 2 = ... = 70 cm/s.
`dt = `ds / vAve = 10 cm / (70 cm/s) = .14 sec approx.
Now analyze horiz. motion:
We know that
`ds = 20 cm
a = 0.
Since a = 0, velocity is unchanging and
vf = v0 = vAve.
We know `dt from the analysis of vertical motion, so we can find the average horizontal velocity:
vAve = `ds / `dt = 20 cm / (.14 sec) = 140 cm/s approx.
The second situation is analyzed in exactly the same manner. The time of fall will be very close to .5 sec and the average horizontal velocity will be close to 120 cm/s.
What was the percent difference in the two average velocities?
The difference is 20 cm/s. The two results have equal standing, so we divide by the average of the two numbers, which is 130 cm/s. We get
A marble rolls 20 cm up a ramp, accelerating at 50 cm/s^2, before it turns around and rolls back down.
What was its intial velocity and how long did it take to come to rest before rolling back down?
We know the following, choosing the positive direction as down the ramp:
`ds = -20 cm (20 cm up the ramp is in the negative direction since down the ramp was chosen to be the positive direction)
a = 50 cm/s^2 (acceleration is down the ramp, which is the positive direction)
vf = 0 (the ball comes to rest at max displacement)
We can use the equation vf^2 = v0^2 + 2 a `ds to find v0.
Solving for v0 we get
v0 = +- sqrt( vf^2 - 2 a `ds) = ... = +-45 cm/s.
The ball was started up the ramp, which is the negative direction, so we discard the positive solution and get
v0 = - 45 cm/s.
We then reason out `dt:
vAve = (vf + v0) / 2 = (0 + (- 45 cm/s) ) / 2 = -22.5 cm/s.
We find `dt = `ds / vAve = -20 cm / -22.5 cm/s = .9 sec, approx.
When was the ball at the position 10 cm from its starting point?
Now we know the following:
The init vel. is -45 cm/s, as we just found.
Accel is still 50 cm/s^2, since it's the same situation as before.
However `ds is not -10 cm, not -20 cm.
As a result `dt and vf will be different than for the phase we calculated before.
So we know v0, a and `ds. We use the fourth equation to find vf and get
vf = +-sqrt( v0^2 + 2 a `ds)
= +- sqrt( (-45 cm/s)^2 + 2 * (-10 cm) * (50 cm/s^2) )
= +- sqrt( 1960 cm^2/s^2 - 1000 cm^2 /s^2)
= +- sqrt(960 cm^2 / s^2)
= +- 31 cm/s, approx..
Both solutions are valid. One corresponds to the first time the ball reaches the -10 cm position, the other to its return as it rolls back down the ramp.
Each solution results in a different vAve and therefore a different `dt.
Testing:
Briefly, go to your homepage at http://vhmthphy.vhcc.edu/ and click on Tests.
Choose your course and click on Major Quiz.
Print off the test.
Sign the danged thing.
Get it signed by the attendant.
Take it in the room to which the attendant directs you.
Hand it to the attendant. </h3>
Experiment:
Set up a ramp system to project a marble off the edge of a table in the horizontal direction and with a constant horizontal velocity.
Observe the horizontal range as the marble falls to the floor.
Observe the horizontal range over a much shorter fall.
Calculate average horizontal velocities for both falls (determine time of fall for each trial and divide time of fall into horizontal range, as shown in Intro Prob Sets and in today's notes).
Write down the equations of uniformly accelerated motion (2 minutes).
Sketch a graph of v vs. t for an object which travels down a 50 cm incline, starting from rest, in 5 seconds. Find the slope and area of the graph.
What would be the units of each of the following: