Exercise Solution

Exercise 1

Place name, date, lab number at top of page. Number the problems and place the letter of the correct answer by each number. For example, if you think the answer to 1 is A and the answer to 2 is B, then you would write

Name:

Date:

Lab number: 1-2

etc.

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1. B
The distance of D from the origin (0) at time = 0 s is greater than the distance of C from the origin. D is displaced from the origin and C is at the origin.
Note: It was pointed out in the chapter (Kinematics in One Dimension) we usually choose the x axis as the direction along which motion takes place.

Figure 2-1

At t = 0 s

A.	Rider C is ahead of rider D.
B.	Rider D is ahead of rider C.
C.	Rider C and D are at the same position.

2. D
They are not moving at t = 0. They are, however, at different locations.
Compare this with throwing 2 balls horizontally. They are initially at rest until a velocity is provided.
The velocities (delta x/delta t) are constant since there is no gravity acting in the x direction.
Note: The answer does not conflict with the answer for problem 3 below.

Figure 2-1

At t = 0 s

A.	C is moving, and D is at rest.
B.	D is moving, and C is at rest.
C.	C and D are both moving.
D.	C and D are both at rest.

3. A
Velocity means instantaneous velocity.
When average velocity is to be used it will be referred to as average velocity.
Also, since instantaneous velocity is defined as the limit of delta x over delta t as delta t approaches 0, the velocity at 0 is greater for C (greater slope) than it is for D.
Do not confuse this with, for example, a car starting at time 0 and accelerating to a certain velocity. The plot of x versus t would not be a straight line as shown here because of the acceleration.
The only problems we have discussed in chapter 2 (one-dimensional kinematics) are those in which
a. An object was dropped with an initial velocity of 0. Velocity changes because of g.
b. An object was thrown down with an initial velocity given. Velocity changes because of g.
c. An object is thrown up - g also acts on the object.
d. An obect is thrown horizontally. There is no g in the x direction so V is constant.

Figure 2-1

At t = 0 s

A.	C has a greater velocity than D.
B.	D has a greater velocity than C.
C.	C and D have the same velocity.
D.	C is accelerating.

Additional Graph
What is Shown Here?

4. A
At t = 10 seconds the plots intersect which means that they are at the same position (x values are equal)

Figure 2-1

At t = 10 s

A.	C and D are at the same position.
B.	C and D have the same velocity.
C.	The velocity of D is greater than the velocity of C.
D.	C is in front of D.

5. A
During the first 8 s: The slope of C is negative (decreasing velocity). The slope of D is positive (increasing velocity)
They cannot have the same velocity or same average velocity since velocity is a vector.

Figure 2-2

During the first 8 s

A.	C has decreasing velocity and D has increasing velocity.
B.	C and D both have decreasing velocities.
C.	C and D have the same velocity.
D.	C has the same average velocity as D.

6. B
During the first 8 s: The accelerations (look at the slope) are clearly equal in magnitude but opposite in sign.
Acceleration is a vector...

Figure 2-2

During the first 8 s

A.	the magnitude of the acceleration of C is greater than the magnitude of D's acceleration.
B.	their accelerations are equal in magnitude but opposite in sign.
C.	their accelerations are equal in magnitude and equal in sign.

7. A
An object at rest is not changing postion as time changes. A is the only 1 for which this is true.
Options B and C are clearly changing position as time increases.
What is the situation in Option B? What is the situation in Option C?
What is wrong with option D?

Which graph represents an object at rest?

A.
B.
C.
D.

Exercise 2

Note the Clear Instructions Given as to How To Submit Responses

Unless told otherwise, assume that air resistance is negligible for all problems.

I derived 4 kinematic equations on my Web site; they were also derived in your text. You can use them, or any appropriate manipulation of them, in your calculations.

Throughout the course, show all of your work for problems that require calculations. If a problem involves reasoning only, then describe the reasoning (logic) you used to arrive at your conclusion.

Added Instructions: Not on your exercise but applicable for all future exercises.

To avoid confusion, place the reasoning or calculations with your response to the problem. Some did not follow instructions above and submitted the original drawings which were ignored. This, in turn, caused me to erroneously omit the rationale, etc of a student that placed it separately. Clear instructions are intended to avoid such difficulties - follow them.

1. Assume constant acceleration for this problem. A car accelerates from 13 m/s to 25 m/s in 6.0 s.

a. What was its acceleration?

By definition, the acceleration is a = (v - v₀₎/t = (25 m/s - 13 m/s)/6.0 s = 2.0 m/s²

b. How far did it travel in this time?

x - x₀ = v₀t + (1/2)at²= 114 m

2. A helicopter is ascending vertically with a speed of 5.20 m/s. At a height of 125 m above the Earth, a package (at this height) is

dropped from the helicopter. How much time does it take for the package to reach the ground?

Choose downward to be the positive direction, and take y₀ = 0 to be the height where the object was released. The initial velocity is

v₀= -5.20 m/s², the acceleration is a = 9.80 m/s², and the displacement of the package will be y = 125 m.

The time to reach the ground can be found from y = y₀ + v₀t + (1/2)at²

Inserting values and placing the above in the form of the quadratic formula and solving for t gives +t = 5.61 s

3. What must be your car's average speed in order to travel 235 km in 3.25 hours?

Dd/Dt 235/3.25 = 72.3 km/h

4. A rolling ball moves from x₁ = 3.4 cm to x₂ = -4.2 cm during the time t₁ = 3.0 s to t₂ = 6.1 s. What is its average velocity?

The average velocity is given by delta x/delta t = (-4.2cm -3.4 cm)/(6.1 s - 3.0 s) = -2.5 cm/s

5. Define the following, as they are used in chapter 2

a. Scalar quantity

Quantities specified completely by a number and units. Examples of scalar quantities are mass, time, and temperature.

b. Vector quantity

A vector quantity is one that has direction as well as magnitude. Examples of vector quantities are velocity, displacement, force, and momentum.

c. Particle

A particle is a mathematical point that has no spacial extent. It can undergo only translational motion.

d. Mechanics

The study of the motion of objects, and the related concepts of force and energy.

e. Kinematics

The description of how objects move.

f. Dynamics

Deals with force and why objects move as they do.

g. Instantaneous acceleration

instantaneous acceleration = lim D v

D t ®0 D t

h. t

time after time 0

i. t₀

initial time

j. g

gravity

6. What is the difference between displacement of an object and the distance travelled by an object?

Distance is the length of the total path travelled by an object.

Displacement is the change in position of the object.

If you travel in a straight line, then they are equal. Otherwise not equal.

7. You travel from point A to point B in a car moving at a constant speed of 70 km/hour. Then you travel the same distance from point B

to another point C, moving at a constant speed of 90 kn/hour. What is your average speed?

The average speed is NOT 80 km/h.

Since the two distances traveled were the same, the times of travel were unequal.

The time to travel from A to B at 70 km/h is longer than the time to travel from B to C at 90 km/h.

Thus we cannot simply average the speed numbers.

To find the average speed, we need to calculate (total distance) / (total time).

We assume the distance from A to B and the distance from B to C are both d km.

The time to travel a distance d with a speed v is t = d / v.

v = (d_av + d_bc)/(t_ab + t_bc) = d km + d km

d km + d km

70 km/hr 90 km/hr

after adding the terms in the numerator, combining the terms in the denominator, inverting the denominator and multiplying

it by the numerator, we arrive at the answer of

78.75 km/hr

8. Which one of these motions is not at constant acceleration: Explain your answer.

a. a rock falling from a cliff

Ignoring air resistance, a rock falling from a cliff would have a constant acceleration. (If air resistance is included, then the acceleration will be decreasing as the rock falls.)

b. an elevator moving from the second floor to the fifth floor making stops along the way

The elevator moving from the second floor to the fifth floor is NOT an example of constant acceleration. The elevator accelerates upward each time it starts to move, and it accelerates downward each time it stops.

c. a dish resting on a table?

A dish resting on a table has an acceleration of 0, so the acceleration is constant.

9. A person throws a ball upward in the air with an initial velocity of 18.0 m/sec. How long is the ball in the air before it comes back to

his hand?

Choose positive as up and place the origin at the person's hand. y will then be 0 at the point the ball is thrown and again when it returns to his hand.

y = y₀ + v₀t + (1/2)at²

y = (18.0 m/s)t + (1/2)(-9.80 m/s²)t² = 0 since y₀is 0

factoring out one t we arrive at

(18.0 m/s - (4.90 m/s²t)t

this equation has the following 2 solutions

t = 0 s, which is when the ball is thrown and

t = 18.0/4.90 = 3.67 s which is when the ball returns to his hand

so, the anwer is 3.67 s

10. A ball is thrown downward at an initial velocity of 4 m/sec from a tall building. What is its velocity after 5 seconds?

Place the coordinate system at point of throwing with positive direction down.

v₀ = 4 m/sec, t = 5 s, a = 9.80 m/s², y₀ = 0

v = v₀+ at v = (4 m/s) + 9.80 m/s²)( 5 s) = 4 m/s + 49 m/s = 53 m/s