Physics 101 Sample Problems with Solutions - Kinematics

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1. An airplane accelerates down a runway at 3.20 m/s2 for 32.8 s until is finally lifts off the ground. Determine the distance traveled before takeoff.
   

   Given: a = +3.2 m/s2 t = 32.8 s vi = 0 m/s

   Find: d = ??

   d = v_i*t + 0.5*a*t²
   d = (0 m/s)*(32.8 s)+ 0.5*(3.20 m/s²)*(32.8 s)²
   d = 1720 m
2. A race car accelerates uniformly from 18.5 m/s to 46.1 m/s in 2.47 seconds. Determine the acceleration of the car and the distance traveled.
   

   Given: vi = 18.5 m/s vf = 46.1 m/s t = 2.47 s

   Find: d = ?? a = ??

   a = (Delta v)/t
   a = (46.1 m/s - 18.5 m/s)/(2.47 s)
   a = 11.2 m/s²
   d = v_i*t + 0.5*a*t²
   d = (18.5 m/s)*(2.47 s)+ 0.5*(11.2 m/s²)*(2.47 s)²
   d = 45.7 m + 34.1 m
   d = 79.8 m
   (Note: the d can also be calculated using the equation v_f² = v_i² + 2*a*d)
3. A feather is dropped on the moon from a height of 1.40 meters. The acceleration of gravity on the moon is 1.67 m/s². Determine the time for the feather to fall to the surface of the moon.
   

   Given: vi = 0 m/s d = -1.40 m a = -1.67 m/s2

   Find: t = ??

   d = v_i*t + 0.5*a*t²
   -1.40 m = (0 m/s)*(t)+ 0.5*(-1.67 m/s²)*(t)²
   -1.40 m = 0+ (-0.835 m/s²)*(t)²
   (-1.40 m)/(-0.835 m/s²) = t²
   1.68 s² = t²
   t = 1.29 s
4. A kangaroo is capable of jumping to a height of 2.62 m. Determine the takeoff speed of the kangaroo.
   

   Given: a = -9.8 m/s2 vf = 0 m/s d = 2.62 m

   Find: vi = ??

   v_f² = v_i² + 2*a*d
   (0 m/s)² = v_i² + 2*(-9.8 m/s²)*(2.62 m)
   0 m²/s² = v_i² - 51.35 m²/s²
   51.35 m²/s² = v_i²
   v_i = 7.17 m/s
5. If Michael Jordan has a vertical leap of 1.29 m, then what is his takeoff speed and his hang time (total time to move upwards to the peak and then return to the ground)?
   

   Given: a = -9.8 m/s2 vf = 0 m/s d = 1.29 m

   Find: vi = ?? t = ??

   v_f² = v_i² + 2*a*d
   (0 m/s)² = v_i² + 2*(-9.8 m/s²)*(1.29 m)
   0 m²/s² = v_i² - 25.28 m²/s²
   25.28 m²/s² = v_i²
   v_i = 5.03 m/s
   To find hang time, find the time to the peak and then double it.
   v_f = v_i + a*t
   0 m/s = 5.03 m/s + (-9.8 m/s²)*t_up
   -5.03 m/s = (-9.8 m/s²)*t_up
   (-5.03 m/s)/(-9.8 m/s²) = t_up
   t_up = 0.513 s
   hang time = 1.03 s

PHYSICS 101AH: LESSON 2

PHYSICS 101AH: LESSON 2

SCALARS AND VECTORS

2.1. General Categories of Physical Quantities

Scalar quantities are those quantities which are completely specified by their magnitude,

expressed in some convenient units.

·         easy to handle since they can be manipulated by ordinary laws of algebra (Usually can be added, subtracted, multiplied, divided directly)

·         Examples: length, mass , area, volume, time, density

Vector quantities are those which require for complete specification, both magnitude and

direction.

·         Direction is just as important as the magnitude when specifying quantities

·         It always represented by an arrow.

·         Examples: displacement, force, acceleration, velocity, momentum

·         The sum of the vectors is called the RESULTANT vector.

2.2. Addition of Vectors

            Methods in Adding Vectors

A.     Graphical Method – also known as Geometrical Method and require no computation.

a.       Parallelogram Method (Tail to Tail Method) – used to add only two vectors

b.      Triangle Method (Head to Tail Method) – use to add only two vectors

c.       Polygon Method (Head to Tail Method) – use to add more than 2 vectors

An illustration will help for Head to Tail method: Note that we can move vectors around as we wish on the graph paper, because only the magnitude and direction matter. The location does not. Also note that this is the way we would add together displacements – if I say "go along vector A and then along B", then in the end we would end up in the same place as if we had just gone along vector R, the resultant. In that sense, R is the sum of A and B. We take this as a general definition of adding any two vectors together, whether they are displacements, velocities, or some other quantities.
 
            To do this accurately, we need to use a ruler and protractor. The general technique for adding two vectors on graph paper is as follows:

1.      Start at the origin and draw the first vector (based on the numbers given you)

2.      At the end of the first vector, make a new origin

3.      Draw the second vector starting at the new origin

4.      Connect the tail of the first vector to the head of the second

When you are finished, the last line you have drawn is the resultant. You can then measure the length and angle of this vector using a protractor and ruler.

B.     Analytical Method (Component Method/Trigonometric Method)

Trigonometry is also important in physics. When you have a right-angled triangle, the following relationships are true:

           

This method is based on the fact that we can specify a vector by specifying its magnitude in two perpendicular directions. We take these to be the x and y directions. We call the length of the vector in the x direction as the x-component of the vector, and similarly for y. The nice thing about this method is that once we have the x and y components of the vectors we want to add, adding them is simple. Let us say that we have two vectors, A and B, with the components labeled as A_x, A_y, B_x, and B_y. It should be obvious which component each symbol stands for. If our resultant, or sum vector, is called R, with components R_x and R_y, then we have

R_x = A_x + B_x                                                R_y = A_y + B_y 

But how do we get the components if we are given the angle and direction? We use trigonometry. Consider the diagram:

We know from trigonometry that
                        A_x = A cos q 
                        A_y = A sin q

Similar relations hold for the components of B, or any other vector for that matter. Here A is the magnitude of A, and q is the angle.

We now need to be able to go back from the components to the magnitude and direction. We have, again from trigonometry,
                                    R² = R_x² + R_y² 
                                    q = tan^-1 (R_y / R_x)  

 The process for adding two vectors, A and B, is thus:

1. Find the components of A and B using (2a) and (2b)
2. Add the components together using (1a) and (1b)
3. Find the magnitude and direction of R using (3a) and (3b)

Note: Calculators are funny things. You should always make a sketch of your addition to see if your values agree with what your calculator gives. The magnitudes should be correct, but the angles will often come out wrong. To adjust the angles, note that

                        cos q = cos (360° - q )

                         sin q = sin (180° - q )

                         tan q = tan (180° + q )

            The summarized steps for component method are as follows:

1.      Resolve all the given vectors into their x and y components.

2.      Find the algebraic sum of all the x-components (∑X), and the algebraic sum of all the y-components (∑Y).

3.      Find the magnitude of the resultant using the Pythagorean theorem.

R = sqrt ((∑X)² + (∑Y)²)

4.      determine the angle of the resultant and the specific direction using the tangent function.                 

θ = arctan (∑Y /  ∑X)

PHYSICS 101AH: LESSON 1

PHYSICS 101AH: LESSON 1

INTRODUCTION TO PHYSICS & SCIENTIFIC MEASUREMENTS

PHYSICS. The word “physics” (Latin, physics, or Greek, physis), means nature and Physics is the study
of the laws that determine the structure of the universe with reference to the matter and energy. It is
concerned not with the chemical changes that occur but with the forces that exists between objects and
the interrelationship between matter and energy (Parker, 1994).

There are six branches of Pgysics: (1) Mechanics: kinematics/ dynamics; (2) Heat & Thermodynamics; (3)
Sound and Waves; (4) Light and Optics; (5) Electricity and Electromagnetism; and (6) Modern Physics.

A measurement can be defined as number with attached units. The numerical value of a measurement
should reflect the sensitivity of the measurement used to make the measurement.

Consider a bathroom scale, which measurement is reasonable? 165.674 lbs or 166 lbs. An ‘exact
measurement’ does not really exist because every instrument has some degree of uncertainty. An
instrument reads only a finite number of digits that have meaning.

Every measurement has some degree of uncertainty in the last decimal place. The last digit read with an
instrument, with analog readout, is estimated.


1.1 Methods: Direct and Indirect

1.2 Quantities: Fundamental (Basic) and Derived

Standards of Length, Mass and Time

There are five basic quantities:

• Length (L)
• Mass (M)
• Electric current (I)
• Temperature (T)

1.3 Units: Fundamental and Derived

1.4 System of Units: English System (fps/fss system) and Metric System/ S.I. (mks/cgs system)

In the first of the course we will only be concerned with length, mass, and time.

SI units (used mostly in physics):

• Length: meter (m)
• Mass: kilogram (kg)
• Time: second (s)

This system is also referred to as the mks system for meter-kilogram-second.

Gaussian Units (used mostly in physics/ chemistry):

• Length: centimeter (cm)
• Mass: gram (g)
• Time: seconds (s)

This system is also referred to as the cgs system for centimeter-gram-second.

British Engineering System:

• Length: foot (ft)
• Mass: slug
• 
  
• Time: second (s)

1.5 Significant Figures: All Non-zero digits

RULES FOR DETERMINING SIGNIFICANT DIGITS

1. Nonzero digits are always significant.

2. Leading zeros that appear at the start of a number are never significant because they act only to
fix position of the decimal point in a number less than 1.

3. Confined zeros that appear between nonzero numbers are always significant.

4. Trailing zeros at the end of a number are significant only if the number contains a decimal point
or contains an over-bar.

1.6 Rules for Rounding Off

1. If the first non-significant digit is less than 5, drop it and the last significant digit remains the

2. If the first non-significant digit is more than 5 or is 5 followed by numbers other than zeros, drop
the non-significant digit(s) in increase the last significant digit by 1. Hence, 47.26 and 47.252 are
both equal to 47.3 (when rounded to 3 sig. figs)

3. If the first non-significant digit is 5 and is followed by zeros, drop the 5 and,
a) increase the last significant digit by one if it is odd, or
b) leave the significant digit the same if it is even.
c. the new rule is hereby accepted too. That when the number to be dropped is exactly five, add
immediately one to the preceding number.

4. Non-significant digits to the left of the decimal point are not discarded, but are replace by zeros.
Thus 1781 becomes 1780 and not 178 when rounded to three significant figures.

1.7 Rule for Addition and Subtraction

The answer must not contain a smaller place than the number with the smallest place.

1.8 Rule for Multiplication and Division

The answer must not contain any more significant digits than the least number of significant digits in
the numbers used in multiplication or division.

1.9 A Special Rule: Exact Numbers

Exact numbers are precisely know and can have as many significant digits as a calculation requires,
so there are not used to determine the number of significant digits for the answer.

Exponential Notation

Form of mathematical expression in which a number is expressed as the product of two numbers,
one a decimal and the other a power of 10. (1000 = 1 X 103)

Form of exponential notation in which the decimal part must have exactly one nonzero digit to the
left of the decimal point; it is widely used by scientist.

Conversion and Equivalencies

Idea: Units can be treated as algebraic quantities. For example, we can use the conversion factor 1in
= 2.54cm to rewrite 15 inches in centimeter.

15 in = 15 in (2.54 cm/ 1 in) = 38.1 cm

• The basic unit of measurement in Metric System is meter.
• On a ruler, the distance from one tick mark to the next is 1 millimeter
• Millimeters are used to measure very short lengths.
• There are 10 millimeters in a centimeter.
• For large distances, you would measure in kilometers.
• All metric units are related by units of tens.
• The basic metric unit of mass is gram
• There are 1000 grams in one kilogram
• The metric ton is used to measure very heavy objects
• The milligram is used to measure very light objects

• It is common outside the field of science to use weight to mean the same as mass.
• In Physics: Mass is different from weight.
• Mass is the measure of the amount of matter in an object.
• Weight is the force exerted by gravity on an object.
• A scale uses a spring to measure weight. A balance measures mass by comparing the force
• acting equally on both pans of a balance.
• Capacity is the term used for measuring a liquid.
• The dictionary defines capacity as the amount of space that can be filled
• The liter is the basic unit of capacity in the metric system
• The milliliter is used to measure very small capacities
• The kiloliter is used to measure large capacities
• There are1000 liters in every kiloliter

Commonly used prefixes for powers of 10 used with metric units are given below:




nano

micro

mili

centi

deci

kilo

mega