Answer:
A
Explanation:
F = m * a
100 = 5 * a
a = 100/5 = 20 m/s^2
Identify the law, write the equation and calculate the answer to the problem below.
An incident light ray strikes water at an angle of 45 degrees. The index of refraction of air is 1.0003, and the index of refraction of water is 1.33. What is the angle of refraction rounded to the nearest whole number?
Find refractive index first
[tex]\\ \rm\Rrightarrow \mu=\dfrac{c}{v}[/tex]
[tex]\\ \rm\Rrightarrow \mu=\dfrac{1.0003}{1.33}[/tex]
[tex]\\ \rm\Rrightarrow \mu =0.75[/tex]
Now
[tex]\\ \rm\Rrightarrow \dfrac{sini}{sinr}=\mu[/tex]
[tex]\\ \rm\Rrightarrow \dfrac{sin45}{sinr}=0.75[/tex]
[tex]\\ \rm\Rrightarrow \dfrac{sin45}{0.75}=sinr[/tex]
[tex]\\ \rm\Rrightarrow sinr=0.94[/tex]
[tex]\\ \rm\Rrightarrow r=sin^{-1}(0.94)[/tex]
[tex]\\ \rm\Rrightarrow r=70^{\circ}[/tex]
Your heart pumps about 70 mL of blood per beat and beats an average of 80 Times per minute how many gallons of blood does your heart pump every hour
Answer:
88.761
Explanation:
R S ( M ) = 2 G M c 2 , where G is the gravitational constant and c is the speed of light. It is okay if you do not follow the details of the equation; the basic point is that if you put an amount of mass M or larger in a sphere of radius smaller than RS(M)
What equation is this
The provided question's answer is "Schwarzschild radius".
The conversion factor between mass and energy is the speed of light squared.
GM/r stands for gravitational potential energy, also known as energy per unit mass.
GM/rc² then has "mass per unit mass" units. In other words, as mass/mass splits out in a dimensional analysis, "dimensionless per unit."
The derivation yields a formula for time or space coordinate ratios requiring sqrt(1 - 2GM/rc²). This number becomes 0 when r=2GM/c2, or the formula becomes infinite if in the denominator. However, there is no justification for using c² as a conversion factor there. Consider the initial expression sqrt(1 - 2GM/rc²).
Assume that m is used as the test particle's mass instead of 1. Then you have sqrt(m - 2GMm/rc² and mass units. This expression denotes that the rest energy of the test mass m you introduced into the gravitational field is "gone" at that radius.
The 2 would be absent if the gravitational field were Newtonian. However, at the event horizon, Einstein gravity is slightly stronger than Newton gravity, resulting in the factor 2 in qualitative terms.
So, the given equation is of Schwarzschild radius.
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Which picture correctly shows the path of refracted light rays given an object outside the focal point? Select one: a. A b. B c. C d. D
Answer:
Answer is C because light travels in a sight line but when light pass through a refractor the light from the source changes direction when passes through a refractor
An ultraviolet wave traveling through a vacuum has wavelength of 4.0 x 10^-7 m. The waves frequency, written in scientific notation to two significant figures, is ? X10^14Hz.
Answer:
λ = c / f or f = c / λ
f = 3.0E8 / 4.0E-7 = .75E15 / sec = 7.5E14 / sec = 7.5 X 10^14 /sec
a generator produces a maximum emf of 240 v when rotating at 3.7 rad/s in a magnetic field of 0.200 t. if the coil has 822 loops what is the area of each loop?
Answer:
I actually did that but I fprgot
Answer: 0.0387
Explanation: I had the same problem and here's what I got:
Values:
ε (Voltage) = 240
ω (Angular velocity) = 37.7
B (Magnitude) = 0.200
N (Number of loops) = 822
ε = NABω (Rearrange to get A)
A = ε/WBN
= (240)/(37.7)(0.200)(822)
A = 0.0387 (FINAL ANSWER)
A 63.3 kg wood board is resting on very smooth ice in the middle of a frozen lake. A 35.7 kg boy stands at one end of the board. He walks from one end of the board to the other end with a velocity of 1.47 m/s relative to the ice in the positive direction. What is the velocity of the board relative to the ice?
The velocity of the board relative to the ice is zero, since both are at rest.
What is relative velocity?Relative velocity is the velocity of an object in relation to another reference object or point.
When two objects are travelling or moving with the same velocity in the same direction, the relative velocity one relative to the other is zero.
Also, when two objects are at rest, the relative velocity one relative to the other is zero.
Therefore, the velocity of the board relative to the ice is zero, since both are at rest.
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Please help me!
If ball C is 3 times the volume of ball D and ball D has 1/3 the mass of ball C, which has the greater density?
A. Ball C
B. Ball D
C. The Densities are equal
C. The Densities are equal.
What is density?
Density is mass per unit volume or mass of a unit volume of a material substance.
If m1, V1 and D1 = mass, volume and density respectively of ball C
m2, V2 and D2 = mass, volume and density respectively of ball D
According to the Question ,
[tex]V_{1} = 3V_{2} , m_{2} = \frac{1}{3} (m_{1} ) \\ \\= m_{1} = 3m_{2}[/tex]
Therefore,
[tex]\frac{D_{1} }{D_{2} } = (\frac{m_{1} }{V_{1} } )* (\frac{m_{2} }{V_{2} } )\\ \\= (\frac{3m_{2} }{3V_{2} })*(\frac{V_{2} }{m_{2} }) \\\\= 1[/tex]
Hence, D1 = D2
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the primary coil of a transformer has 480 turns, and the secondary has 112 turns. when an ac voltage of 28.0 v is applied to the primary, what is the output voltage in the secondary?
The voltage of the secondary will be 36 V.From the given conditions primary transformer has 3 times as many turns in the secondary coil.
What is induced voltage?Electromagnetic induction is what causes the induced voltage. Electromagnetic induction is the process of generating emf (induced voltage) by subjecting a conductor to a magnetic field.
Given data;
No turns in the primary coil,[tex]\rm N_p = 480 turns[/tex]
No turns in the secondary coil,[tex]\rm N_s = 112 turns[/tex]
The voltage of the primary coil,[tex]\rm V_p = 28 v[/tex]
The voltage of the secondary coil,[tex]\rm V_s = ?[/tex]
For a transformer,
[tex]\rm \frac{V_p}{V_s}= \frac{N_p}{N_s} \\\\ \rm \frac{28}{V_s}= \frac{480}{112} \\\\ V_s = 6.53 \ v[/tex]
Hence the voltage of the secondary will be 36 V.
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Answer: 6.53
Explanation: I had the same question and here's what I got:
Values:
Vp (Primary Voltage) = 28.0
Ns (Secondary Number of Turns) = 112
Np (Primary Number of Turns) = 480
Vs = Vp Ns/Np
= (28)(112)/(480)
Vs = 6.53 (FINAL ANSWER)
water
0.6m
water wave
tank
How long does it take for the wave to return to the
position XY, but moving to the right?
[3]
b A man is cutting down a tree with an axe. He
hears the echo of the impact of the axe hitting
the tree after 1.6 s.
i What sort of obstacle could have caused the
echo?
ii The speed of sound is 330 m/s. How far is
the tree from the obstacle?
c Distinguish between the nature of the sound
wave in b and the water wave in a.
[2]
ii the amplitude of
b The cone of a louds
diagram shows how
out in front of the c
loudspeaker
P is a compression,
i
Describe how
changes from
ii Describe the
the sound w
iii Copy the di
and mark a
wavelength
5 a The first diagrams
Answer:
Discrimination is the worst thing in the world you can't even do a thing so you will do such a physical thing or do a mathematics problem ok done it's ok
Which picture shows the wave with the smallest wavelength?
Select one:
a. A
b. B
c. C
d. D
Answer:
b. B
Explanation:
Picture B has the smallest peaks among all which henceforth makes the wavelength i.e. distance between two adjacent crests or troughs, small.
Answer:b
Explanation:Wavelength can be calculated using the following formula: wavelength = wave velocity/frequency. Wavelength usually is expressed in units of meters. The symbol for wavelength is the Greek lambda λ, so λ = v/f.
(the problem setup and question are below)
The gauge pressure at bottom of vaccine solution will be 16 kPa
Positive pressure is another name for gauge pressure. When a system's internal pressure exceeds that of its surroundings, it is said to be under positive pressure. Any leak that develops in the positively pressured system will therefore escape into the outside world. In contrast, a negative pressure chamber draws air into it.
Given As seen in the illustration, a syringe is held vertically. The container carries a 3 cm tall column of vaccine solution and has an open inner diameter of 1 cm. The needle contains a 2 cm column of vaccine solution and has an open inner diameter of 0.5 mm. At the needle's open end, the solution is exposed to the air. The vaccination solution has a density of 1200 kg/m3.
We have to find the gauge pressure at bottom of vaccine solution
Since the 5N force is applied to vaccine solution the pressure exerted will be much more
Hence the gauge pressure at bottom of vaccine solution will be 16 kPa
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A solenoid with 465 turns has a length of 8.00 cm and a cross-sectional area of 3.10 ✕ 10−9 m2. Find the solenoid's inductance and the average emf around the solenoid if the current changes from +3.50 A to −3.50 A in 7.83 ✕ 10−3 s
The solenoid's inductance and the average emf around the solenoid if the current changes from +3.50 A to −3.50 A in 7.83 ✕ 10−3 s will be 1.01 ×10⁻⁸ H and 9.02 × 10⁻² V.
What is a solenoid?A coil of wire that carries an electric current is a solenoid. A solenoid is an electromagnet formed by a helical coil of wire.
Which generates a magnetic field when an electric current is passed through the coil.
A solenoid is a form of coil that produces a magnetic field when the electric current is passed through it. A solenoid is created when a conductive wire is used to make a loop.
Given data;
Turns ,N = 465
Length,L= 8.00 cm
Cross-sectional area,A = 3.10 ✕ 10−9 m2.
Solenoid's inductance, L=?
The average emf around the solenoid, E=?
Time of flow,t= 7.83 ✕ 10−3 s
Current changes from +3.50 A to −3.50 A
The inductance of the solenoid is found as;
[tex]\rm L = \frac{\mu_0 AN^2}{L} \\\\ \rm L = \frac{4 \times \pi \times 10^{-7}\times 3.0 \times 10^{-9} \times (465)^2 }{8.00 \times 10^{-2}} \\\\ L= 1.01 \times 10^{-8} \ H[/tex]
The average emf around the solenoid is found as;
[tex]\rm e = L \frac{I_2-I_1}{t} \\\\ \rm e = 1.01 \times 10^{-8} \times \frac{3.50-(-3.50)}{7.83 \times 10^{-3}} \\\\ e =9.02 \times 10^{-12} \ V[/tex]
Hence, the solenoid's inductance will be 1.01× 10⁻⁸ H.
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Question 2 of 25
What is the term for a pair of forces described by Newton's third law?
A.
Normal-friction pair
B.
Action-reaction pair
C.
Kinetic-static pair
D.
Inertia-movement pair
The term for a pair of forces described by Newton's third law is action-reaction pair. Details about Newton's law can be found below.
What does Newton's third law state?Newton's third law of motion states that for every action, there is an equal and opposite reaction.
This law proposed that when an object 1 acts on object 2 with a force of a particular magnitude, object 2 also acts on object 1 with an opposite force of same magnitude.
Therefore, it can be said that the term for a pair of forces described by Newton's third law is action-reaction pair.
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'The wave' is a particular type of pulse that can propagate through a large crowd gathered at a sports arena to watch a soccer, hockey, or CFL game. The elemets of the medium are the spectators, with zero position corresponding to their being seated and maximum position corresponding to their standing and raising their arms. When a large fraction of the spectators participate in the wave motion, a somewhat stable pulse shape can develop. The wave speed depends on people's reaction time, which is typically on the order of 0,1s. Estimate the order of magnitude, in minutes, of the time required for such a pulse to make one circuit around BC Place Stadium in Vancouver.
State all the assumptions that you've made.
Information about BC Place: dimensions are approximately 100 m X 85 m.
The total time required is 1 minute and 0.3 seconds.
Assumption: The distance between the people is 1 m and the stadium is a circle with a radius of 100m.
Here, the time taken by the person is 0.1 seconds.
Total distance covered by the wave = Circumference of the circle
So, total distance = 2 πr = 2 × 3.14 × 100 = 618 m
As the distance between each of the people is 1 m.
So, the number of personal interactions is 618.
Time taken by each person is 0.1 seconds.
So, total time, t = 0.1 × 618 = 61.8 seconds.
So, the order of the magnitude of the time required is 1 minute and 0.3 seconds for such a pulse to make one circuit around BC Place Stadium in Vancouver.
Hence, the total time required is 1 minute and 0.3 seconds.
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A skier (m=55.0 kg) starts sliding down from the top of a ski jump with negligible friction and takes off horizontally.
H = 1.15x10^1 m, h= 6.50 m, and D = 11.4 m.
Find her total kinetic energy as she reaches the ground.
The total kinetic energy of the woman when she reaches the ground is 6,198.5 J.
Total kinetic energy of the woman at the ground levelThe total kinetic energy of the woman when she reaches the ground is calculated from the principle of conservation of energy.
P.E(top) = K.E(bottom)
P.E(top) = mgh
P.E(top) = 55 x 9.8 x 11.5
P.E(top) = 6,198.5 J
Thus, the total kinetic energy of the woman when she reaches the ground is 6,198.5 J.
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The action force is the ballon pushing the air out. What is the magnitude of the reaction force of the air pushing the ballon
The magnitude of the reaction force of the air pushing the balloon would be equal and opposite.
How is the reaction force equal and opposite?A push or a pull that an object experiences as a result of interacting with another item is known as a reaction force.
Newton's third law is officially expressed as follows: There is an equal and opposite reaction to every action. The implication of the statement is that there are always two forces acting on the two interacting objects. The force acting on the first object is equal in size to the force acting on the second. The force acting on the first object is acting in the opposite direction to the force acting on the second object. Force pairs—equal and opposing action-reaction force pairs—always exist in pairs.
Knowing that everything has an equal and opposite reaction according to Newton's second law. The preasured air that a balloon had to push out into the free air acts as the reaction force.
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What method could I use to test this hypothesis? If the mass and the volume of and object are known, then its density can be calculated dividing the object's mass by its volume.
Answer:
The scientific method
Explanation:
A 17-kg sled is being pulled along the horizontal snow-covered ground by a horizontal force of 31 N. Starting from rest, the sled attains a speed of 2.3 m/s in 9.2 m. Find the coefficient of kinetic friction between the runners of the sled and the snow.
What is the number and units?
The sled accelerates with magnitude [tex]a[/tex] such that
[tex]\left(2.3\dfrac{\rm m}{\rm s}\right)^2 = 2a(9.2\,\mathrm m) \implies a = 0.2875\dfrac{\rm m}{\mathrm s^2}[/tex]
By Newton's second law, the net force in the plane of motion (parallel to the ground) is
[tex]31\,\mathrm N - F_{\rm friction} = (17 \,\mathrm{kg}) \left(0.2875 \dfrac{\rm m}{\mathrm s^2}\right)[/tex]
so that the force of friction exerts a magnitude of
[tex]F_{\rm friction} = 26.1125 \,\mathrm N[/tex]
Perpendicular to the ground, the sled is in equilibrium, so Newton's second law says
[tex]F_{\rm normal} - (17\,\mathrm{kg})g = 0 \implies F_{\rm normal} = 166.6 \,\mathrm N[/tex]
The magnitude of friction is proportional to the magnitude of the normal force by a factor of [tex]\mu_k[/tex], the coefficient of kinetic friction. It follows that
[tex]F_{\rm friction} = \mu_k F_{\rm normal} \implies \mu_k = \dfrac{26.1125\,\rm N}{166.6\,\rm N} \approx \boxed{0.16}[/tex]
(and the coefficient is dimensionless).
Which graph shows the correct relationship between kinetic energy and speed? A. Graph representing the relationship between speed on the x-axis and kinetic energy on the y-axis. A straight line starts at the y-axis B. Graph representing the relationship between speed on the x-axis and kinetic energy on the y-axis. A straight line starts from the origin C. Graph representing the relationship between speed on the x-axis and kinetic energy on the y-axis. A curve line runs parallel to the x-axis and y-axis D. Graph representing a relationship between speed on the x-axis and kinetic energy on the y-axis. A semi-curve line starts above the origin on the y-axis and curves upwards as it moves forward in speed
The proper connection between kinetic energy and speed is depicted in Graph D.
What is kinetic energy?The kinetic energy (KE) is defined as one-half of the mass times multiplied by the square of velocity.
[tex]\rm KE = \frac{1}{2}mv^2[/tex]
where,
KE is the kinetic energy
m is the mass of each molecule
(V) is the velocity
As the square of the velocity is exactly proportional to kinetic energy. Consequently, the relationship between velocity and kinetic energy must be parabolic.
The proper connection between kinetic energy and speed is depicted in Graph D. Graph showing the link between kinetic energy on the y-axis and speed on the x-axis.
On the y-axis, a semi-curve line begins above the origin and ascends as it accelerates.
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A contestant in a winter games event pushes a 54.0 kg block of ice across a frozen lake as shown in Figure 4.29(a). The coefficient of static friction is 0.1 and the coefficient of kinetic friction is 0.03.
a) Calculate the minimum force F he must exert to get the block moving.
N
(b) What is its acceleration once it starts to move, if that force is maintained?
m/s2
Answer:
54.0 x0.1=5.4 x0.03=0.162
kinetic force
If a contestant in a winter games event pushes a 54.0 kg block of ice across a frozen lake. The coefficient of static friction is 0.1 and the coefficient of kinetic friction is 0.03, then the minimum force F he must exert to get the block moving would be 5.2974 Nwrons.
What is friction?
Friction is a type of force that resists or prevents the relative motion of two physical objects when their surfaces come in contact.
As given in the problem, If a contestant in a winter games event pushes a 54.0 kg block of ice across a frozen lake. The coefficient of static friction is 0.1 and the coefficient of kinetic friction is 0.03,
The force required to get the block moving = μMg
= 0.01×54×9.81
= 5.2974 Newtons
Thus, the minimum force required to move the block would be 5.2974 Newtons.
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Calculate the estimated density of each ball. Use the formula
D = m/V
where D is the density, m is the mass, and V is the volume. Record your calculations in Table A of your Student Guide.
Given that the density of water is 1.0 g/cm3, make a prediction about whether each ball will float in water. Record your prediction in Table A.
What is the estimated density of the table tennis ball? Record your answer to the nearest hundredth.
0.07
g/cm3
What is the estimated density of the golf ball? Record your answer to the nearest hundreth.
The estimated density of the golf ball is 700 kg/m³
What is density?Density is defined as Mass per unit Volume.
In displacement method,
First , we measuring the volume of water displaced by an object which tell us the volume of the object then we will use the physical balance to determine its mass.
Then calculate the density by dividing the mass by the volume.
i.e. D = m/V
Given, Density of water is 1.0 g/cm³
Using displacement method , The estimated volume of golf ball is 100 cm³ and estimated mass is 7g
Then ,
Density = 10 cm³ / 7 g= 0.07 g/ cm³ = 700 kg/m³
So the estimated density of golf ball is 700 kg/m³
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When passing in a medium for a distance of 1.5 cm intensity
of the light decreased by 3 times. What will the distance x equal to when the intensity of the light decreases by 9 times?
The distance x equal to 0.87 cm when the intensity of the light decreases by 9 times.
Intensity of light
The intensity of light is given as power emitted by the light by unit area.
I = P/A
I = P/L²
I₁L₁² = I₂L₂²
L₂² = I₁L₁²/I₂
L₂² = (3 x 1.5²)/(9)
L₂² = 0.75
L₂ = 0.87 cm
Thus, the distance x equal to 0.87 cm when the intensity of the light decreases by 9 times.
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Determine the amount of power
used in holding a 25 kg box, 1.5
meters above the floor, for 60
seconds.
[?] W
(answer is not 6.13)
Thank you in advance!
Here is your answer mate,
Question,
[tex]Determine\: the\: amount\\ \: of\: power\:used\: in\\\: holding\: a\: 25\: kg\: box\:\\ , \: 1.5\: meters \: above\: the\: floor\\\: for\: 60\: seconds[/tex]
Answer,
Power is equal to work done per unit timeWork is force × displacement SI UNIT OF WORK Newton meterSI UNIT OF POWER Watt[tex][/tex]
Solution,
[tex][/tex]
Given,
[tex]MASS \: IS\: 25\: KG\: \\ and \: HEIGHTIS\: 1.5m\: [/tex]
[tex][/tex]
WORK DONE (done against gravity) =
mass×acceleration due to gravity ×height
WORK = 25× 10× 1.5
[tex]\: \: \: \: \: \: \: \:\: \: \: \: \: \: \: [/tex]= 375 Nm
[tex][/tex]
Now
POWER =
[tex] \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:\frac{work}{time} [/tex]
POWER
[tex]\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \:= \frac{375}{60} Watt [/tex]
[tex]\: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: \: =6.25[/tex]
[tex]Therfore\: your \: answer\: is\: 6.25[/tex]
[tex][/tex]
Check this,
[tex]Acceleration\: due\: to \: gravity\\\: can\: be\: 9.8\: m/s²\: \\As\: nothing\: mentioned\\\: in\: question\: \\I \: took \: it \: as \: 10[/tex]
[tex][/tex]
Have a good day
Calculate the approximate number of atoms in a bacterium, assuming the average mass of an atom is ten times the mass of a hydrogen atom. The mass of a bacterium is 10−15 kg and the mass of a hydrogen atom is of the order of 10−27 kg.
atoms
10¹¹ is the approximate number of atoms in a bacterium.
What do you understand by mass of element?
The atomic mass of an element is the average mass of the atoms of an element measured in atomic mass unit (amu).
Given,
Mass of a bacterium atom = 10⁻¹⁵ kg.
Mass of a hydrogen atom = 10⁻²⁷ kg.
From the above observation ,
The average mass of an atom of the bacterium is ten times the mass of a hydrogen atom.
Atomic mass 1 bacterium atom = 10 x mass of hydrogen atom
= 10 x 10⁻²⁷ kg.
= 10⁻²⁶ kg.
Thus,
The number of atoms in a bacterium
= [tex]\frac{Total mass}{Atomicmass of 1 bacterium}[/tex]
= [tex]\frac{10^{-15} }{10^{-26} }[/tex]
= [tex]10^{11}[/tex]
The approximate number of atoms in a bacterium is [tex]10^{11}[/tex].
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After watching the video below and based on your personal experiences, is there a difference
between areas in precision of control? Could there be differences between left and right cortex
based on experience such as handedness or specific skills such as playing a guitar?
Based on my personal experiences, I believe that there is a difference between areas in precision of control because there are different part of the brain that controls the functions of the body.
What is the difference between left-handed and right-handed people?From the standpoint of brain lateralization, differences do exist such as based on experience such as handedness or specific skills such as playing a guitar.
Note that Left-handers are said to have reduced or little lateralized brains, which tells us that the two halves of the brain are little different than as seen in the right-handers.
Therefore, I can say that based on my personal experiences, I believe that there is a difference between areas in precision of control because there are different part of the brain that controls the functions of the body.
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Charlene has just begun to be able to form a mental representation of an object that is not visibly
present According to Piaget, that means that she has made the transition from the
stage
to the
stage
According to Piaget, that means that she has made the transition from the sensorimotor to preoperational stage
What is the sensorimotor stage?The first of Piaget's four stages of cognitive development is the sensorimotor stage.
A child's understanding that the outside world exists apart from them is what distinguishes it.
Within Piaget's stages of development, the kid will advance to the following stage after they have completely grasped this.
"Charlene has just begun to be able to form a mental representation of an object that is not visibly present.
According to Piaget, that means that she has made the transition from the sensorimotor to preoperational stage
Only recently has Charlene been able to create a mental image of something that is not physically present.
That indicates that she has moved from the sensorimotor to the preoperational level, in accordance with Piaget.
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A block of mass 4 kg is attached to a spring of force constant (k) 1000 N/m, and undergoes simple harmonic motion. What is the period of the motion?
a. 0.1 s
b. 0.4 s
с. 0.2 s
d. 0.3 s
Answer:
0.4
Explanation:
ω = km
ω = 1000 x 4
ω = 4000 divide this by 10000 and you get 0.4s
What is the acceleration of the object?
m/s²
Answer:
-2.5m/s^2
Explanation:
10-40/12-0=-2.5
4. A ball is thrown with an initial speed vi at an angle θi with the horizontal. The
horizontal range of the ball is R, and the ball reaches a maximum height R
6
. In
terms of R and g, find (a) the time interval during which the ball is in motion,
(b) the ball’s speed at the peak of its path, (c) the initial vertical component of
its velocity, (d) its initial speed, and (e) the angle θi
Disclaimer: I ended up finding what's asked for in the reverse order (e)-(a).
At time [tex]t[/tex], the horizontal position [tex]x[/tex] and vertical position [tex]y[/tex] of the ball are given respectively by
[tex]x = v_i \cos(\theta_i) t[/tex]
[tex]y = v_i \sin(\theta_i) t - \dfrac g2 t^2[/tex]
and the horizontal velocity [tex]v_x[/tex] and vertical velocity [tex]v_y[/tex] are
[tex]v_x = v_i \cos(\theta_i)[/tex]
[tex]v_y = v_i \sin(\theta_i) - gt[/tex]
The ball reaches its maximum height with [tex]v_y=0[/tex]. At this point, the ball has zero vertical velocity. This happens when
[tex]v_i \sin(\theta_i) - gt = 0 \implies t = \dfrac{v_i \sin(\theta_i)}g[/tex]
which means
[tex]y = \dfrac R6 = v_i \sin(\theta_i) \times \dfrac{v_i \sin(\theta_i)}g - \dfrac g2 \left(\dfrac{v_i \sin(\theta_i)}g\right)^2 \\\\ \implies R = \dfrac{6{v_i}^2 \sin^2(\theta_i)}g - \dfrac{3{v_i}^2 \sin^2(\theta_i)}g \\\\ \implies R = \dfrac{3{v_i}^2 \sin^2(\theta_i)}g[/tex]
At the same time, the ball will have traveled half its horizontal range, so
[tex]x = \dfrac R2 = v_i \cos(\theta_i) \times \dfrac{v_i \sin(\theta_i)}g \\\\ \implies R = \dfrac{2{v_i}^2 \cos(\theta_i) \sin(\theta_i)}g[/tex]
Solve for [tex]v_i[/tex] and [tex]\theta_i[/tex] :
[tex]\dfrac{3{v_i}^2 \sin^2(\theta_i)}g = \dfrac{2{v_i}^2 \cos(\theta_i) \sin(\theta_i)}g \\\\ \implies 3 \sin^2(\theta_i) = 2 \cos(\theta_i) \sin(\theta_i) \\\\ \sin(\theta_i) (3\sin(\theta_i) - 2 \cos(\theta_i)) = 0[/tex]
Since [tex]0^\circ<\theta_i<90^\circ[/tex], we cannot have [tex]\sin(\theta_i)=0[/tex], so we're left with (e)
[tex]3 \sin(\theta_i) - 2\cos(\theta_i) = 0 \\\\ \implies 3 \sin(\theta_i) = 2\cos(\theta_i) \\\\ \implies \tan(\theta_i) = \dfrac23 \\\\ \implies \boxed{\theta_i = \tan^{-1}\left(\dfrac23\right) \approx 33.7^\circ}[/tex]
Now,
[tex]\cos\left(\tan^{-1}\left(\dfrac23\right)\right) = \dfrac3{\sqrt{13}}[/tex]
[tex]\sin\left(\tan^{-1}\left(\dfrac23\right)\right) = \dfrac2{\sqrt{13}}[/tex]
so it follows that (d)
[tex]R = \dfrac{2{v_i}^2 \times\frac3{\sqrt{13}} \times \frac2{\sqrt{13}}}g \\\\ \implies {v_i}^2 = \dfrac{13Rg}{12} \\\\ \implies \boxed{v_i = \sqrt{\dfrac{13Rg}{12}}}[/tex]
Knowing the initial speed and angle, the initial vertical component of velocity is (c)
[tex]v_y = \sqrt{\dfrac{13Rg}{12}} \sin\left(\tan^{-1}\left(\dfrac23\right)\right) \\\\ \implies v_y = \sqrt{\dfrac{13Rg}{12}} \times \dfrac2{\sqrt{13}} \\\\ \implies \boxed{v_y = \sqrt{\dfrac{Rg}3}}[/tex]
We mentioned earlier that the vertical velocity is zero at maximum height, so the speed of the ball is entirely determined by the horizontal component. (b)
[tex]v_x = \sqrt{\dfrac{13Rg}{12}} \times \dfrac3{\sqrt{13}} \\\\ \implies v_x = \dfrac{\sqrt{3Rg}}{2}[/tex]
Then with [tex]v_y=0[/tex], the ball's speed [tex]v[/tex] is
[tex]v = \sqrt{{v_x}^2 + {v_y}^2} \\\\ \implies v = v_x \\\\ \implies \boxed{v = \dfrac{\sqrt{3Rg}}2}[/tex]
Finally, in the work leading up to part (e), we showed the time to maximum height is
[tex]t = \dfrac{v_i \sin(\theta_i)}g[/tex]
but this is just half the total time the ball spends in the air. The total airtime is then
[tex]2t = \dfrac{2 \times \sqrt{\frac{13Rg}{12}} \times \frac2{\sqrt{13}}}g \\\\ \implies 2t = 2\sqrt{\dfrac R{3g}}[/tex]
and the ball is in the air over the interval (a)
[tex]\boxed{0 < t < 2\sqrt{\frac R{3g}}}[/tex]
