Problem Number 1

Practice Major Quiz.

The Problems follow, with solutions below. You should work the problems, then study the given solutions and compare.

A q_a_ version of the Practice Major Quiz is available on the Supervised Study Current Semester web under Course Documents > Downloads > Physics I.

Problem Number 1

If at clock times t = 0 s, 8 s and 18 s a horse has velocities 10.5 m/s, 10 m/s and 3.25 m/s, then during each of the two time intervals, 0- 8 seconds and 8 - 18 seconds, approximately how far does it move and at what average rate does its velocity change?

Directly reason out your results using the concept of rate.

If the automobile is coasting, and if air resistance is not a significant factor, then over which time interval do you think the average slope of the road is greatest?

Problem Number 2

A straight ramp is inclined at various slopes.

The differences in elevation between one end and the other, for the different slopes, are 1.9, 3.6 and 6.1 cm.
The time required for a cart to coast 78 cm down the ramp, starting from rest, is 2.673299 seconds on the first incline, 2.601031 seconds on the next, and 2.483171 seconds on the last.

How well do the data support the hypothesis that a graph of average acceleration vs. slope will be linear?

Problem Number 3

A projectile leaves the edge of a table and, while traveling horizontally at a constant 31 cm/s, falls freely a distance of 58 cm to the floor. If its vertical acceleration is 980 cm/s², how long does it take to fall and how far does it travel in the horizontal direction during the fall?

Problem Number 4

An object which accelerates uniformly from rest, ending up with velocity 4.2 cm/sec after traveling a distance of 36 cm from start to finish. What are the average acceleration and final velocity of the object?

Problem Number 5

At clock time t = 4 sec, a ball rolling straight down a hill is moving at 5 m/s and is 78 m from the top of the hill, while at clock time t = 9 sec it is moving at 7.5 m/s and is 114 m from the top of the hill.

What is its average velocity during this time? What is its average acceleration during this time? Is it possible that the acceleration is uniform?

Problem Number 1

Directly reason out your results using the concept of rate.

Direct reasoning on 1st interval:

v0 = 10.5 m/s, vf = 10 m/s so

`dv = -.5 m/s

vAve = 10.25 m/s.

Therefore

`ds = vAve * `dt = 10.25 m/s * 8 s = 82 m, approx.

aAve = `dv / `dt = -.5 m/s / (8 s) = -.0625 m/s.

The graph also illustrates the direct reasoning.

Show how you could use a graph of velocity vs. time to obtain your results.

Ave vel is approximated by ave height of trapezoid, change in vel by change in height, time interval by width.

Ave accel is represented by the slope of the trapezoid, displacement by area.

If the automobile is coasting, and if air resistance is not a significant factor, then over which time interval do you think the average slope of the road is greatest?

Greater road slope implies greater magnitude of acceleration. Slope is greatest on the 2d trapezoid, where the greater magnitude of the graph slope implies greater acceleration.

Problem Number 2

A straight ramp is inclined at various slopes.

The differences in elevation between one end and the other, for the different slopes, are 1.9, 3.6 and 6.1 cm.
The time required for a cart to coast 78 cm down the ramp, starting from rest, is 2.673299 seconds on the first incline, 2.601031 seconds on the next, and 2.483171 seconds on the last.

How well do the data support the hypothesis that a graph of average acceleration vs. slope will be linear?

Ramp slopes are 1.9 cm / (78 cm) = .024, 3.6 cm / (78 cm) = .046 and 6.1 cm / (78 cm) = .078.

Accelerations can be calculated from v0=0, `ds =78 cm and the given `dt.

We get accelerations of

21.9 cm/s, 23.1 cm/s, 25.4 cm/s.

Problem Number 3

The vertical motion is independent of the horizontal motion. We analyze the vertical motion first.

Initial vertical velocity is zero. So we know that for vertical motion (motion in the y direction)

v0y = 0, ay = 980 cm/s^2, `dsy = 58 cm.

Using the equations of motion we find that the time interval for the fall is .34 sec.

For basic projectile problems we ignore air resistance, to that horizontal motion is characterized by zero net force and zero acceleration. Therefore, horiz motion is at constant velocity.

The given horiz velocity is 31 cm/s so the horizontal displacement during the fall will be

`dsx = vx * `dt = 31 cm/s * .34 sec = 11 cm, approx.

Problem Number 4

We know v0, vf and `ds. So we use a flow diagram or direct reasoning to conclude that

vAve = 2.1 cm/s (average v0 and vf)
`dt = 36 cm / (2.1 cm/s) = 17 sec approx.
a = `dv / `dt = 4.2 cm/s / (17 s) = .25 cm/s^2.

Problem Number 5

What is its average velocity during this time? What is its average acceleration during this time? Is it possible that the acceleration is uniform?

Between these two events we have `ds = 114 m - 78 m = 36 m and

`dt = 9 s - 4 s = 5 s so

vAve = `ds / `dt = 36 m / (5 s) = 7.2 m/s.

We also know that v0 = 5 m/s and vf = 7.5 m/s. The average of these two velocities is (5 m/s + 7.5 m/s) / 2 = 6.25 m/s. This is not equal to the average velocity.

If accel is uniform we know that vAve = (vf + v0) / 2. Since this is not the case the acceleration ain't uniform.