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Tutor profile: Shubham A.

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Shubham A.
Tutor for 14 years
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Questions

Subject: Physics (Newtonian Mechanics)

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Question:

A car weighing 1500 kg is moving on a highway at a velocity of 60 m/s. The driver notices a speed limit sign of 30 m/s at some distance ahead, and immediately deaccelerates, bringing the speed under the limit within 5 seconds. How much force do the brakes apply on the car? Which law would you use to calculate the force?

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Shubham A.
Answer:

Given that the car deaccelerates, it is bound to be an action of force which is provided by the brakes in this case. As per Newton's second law of motion, the rate of change of momentum of a body is directly proportional to the force applied on the body and takes place in the direction of the force. Thus the force can be easily calculated by figuring out the change in momentum of the car and the rate at which it occurs. Given that: Mass of the car, $$m = 1500$$ kg Initial velocity of the car, $$u = 60$$ m/s Final velocity of the car, $$v = 30$$ m/s Time taken to reduce the velocity, $$\Delta t = 5$$ seconds Determining all the parameters: Initial momentum of the car, $$p_i = m \times u = 1500\times60 = 90,000$$ kg.m/s Final momentum of the car, $$p_f = m \times v = 1500\times30 = 45,000$$ kg.m/s Change in momentum, $$\Delta p = p_f - p_i$$ $$ \Delta p= 45,000-90,000 = -45,000$$ kg.m/s $$ \Delta p= -45,000$$ kg.m/s Now, as per the second law of motion: Force applied by the brakes, $$F = \Delta p/\Delta t$$ $$F =-45,000/5$$ $$F =-9000$$ N Therefore, the brakes apply a retarding force of 9000 N in the direction opposite to the motion of the car. As evident, the force was calculated using Newton's second law of motion.

Subject: Basic Math

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Question:

Does the below system of linear equations have a solution? If so, find it. $$4x-7y = 30$$ $$10x+14y = 12$$

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Shubham A.
Answer:

Before we set out to solve the problem, it is important to note that a lot of information about the nature of the solution can be gathered from the ratios of the coefficients appearing in the equations. It can save precious time and effort if we know in advance whether the solution even exists. For a system of linear equations, there exists a unique solution if the ratio of corresponding coefficients is not equal. Going by the rule, for the equations below: $$4x-7y = 30$$ -(i) $$10x+14y = 12$$ -(ii) We observe the ratios as below: $$\frac{a_1}{a_2} = \frac{4}{10}= \frac{2}{5}$$ $$\frac{b_1}{b_2} = \frac{-7}{14}= \frac{-1}{2}$$ We notice that: $$\frac{a_1}{a_2} \neq \frac{b_1}{b_2}$$ Hence, there exists a unique solution for the given system of equations. We can now obtain this solution using the elimination method, wherein we eliminate one of the variables from the equations. We begin this by multiplying equation (i) by 2: $$(4x-7y = 30)$$ $$\space \times 2 $$ $$8x-14y = 60$$ -(iii) Next, we add equation (ii) and (iii): $$10x+14y = 12$$ $$8x-14y = 60$$ ------------------------- $$18x = 72$$ $$x = 72/18$$ $$x = 4$$ Thus $$x = 4$$. Substituting this value of in equation (i) to find $$y$$: $$4(4)-7y = 30$$ $$16-7y = 30$$ $$-7y = 30-16$$ $$-7y = 14$$ $$y = -2$$ Therefore the solution of the given system of equations is $$x = 4$$ and $$y = -2$$.

Subject: Physics

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Question:

What is light? Given that the speed of light in vaccum is $$3\times10^8$$ m/s and the distance between the earth and the sun is $$1.5\times10^{11}$$m, calculate the time it takes for light from the sun to reach the earth.

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Shubham A.
Answer:

Light is a form of energy that enables us to see. It is an electromagnetic wave, a part of a vast spectrum of electromagnetic radiation, including gamma rays, the X-rays, and the Infrared, among others. As simple and wholesome as this primitive definition is, the story of our understanding of light has been long and one with several twists and turns, a tug of war of sorts. The first rebellious attempt to explain light came from the hallowed Sir Isaac Newton in the 17th century, who gave his famous $$\textbf{corpuscular theory of light}$$, proposing that the light was made up of tiny particles he called corpuscles, and the theory elegantly explained the phenomena of reflection and refraction of light. However soon the theory began to experience chinks, as it was found unable to explain phenomena where the light behaved unlike a particle, essentially diffraction and interference of light. The fact that light seemed to behave as a wave during these phenomena, established the $$\textbf{wave theory of light}$$, first proposed by Huygens. It overruled the authority that Newton enjoyed at the time solely due to the wave theory working better, a beautiful example of how science works as better evidence topples the best of theories by the best of scientists. By the 20th century, the scientists were convinced by the wave nature of light, with the debate over the nature of light largely a thing of past, until Albert Einstein (can we talk Physics without him?), invoked the particle behavior of light to explain the mysterious $$\textbf{photoelctric effect}$$ . With efforts from him and Louis de Broglie, it was later established that light manifested itself as a wave as well as a particle thus showcasing dual behavior, and hence the $$\textbf{wave-particle duality}$$. Light is thus an electromagnetic wave, and unlike mechanical waves as sound, does not need a material medium to travel. Coming to the numerical problem, given that: Speed of light in vacuum, $$c$$ = $$3\times10^8$$ m/s Distance between the sun and the earth, $$s$$ = $$1.5\times10^{11}$$m Hence, time taken by light to reach the earth, $$t$$ = $$\frac{distance}{speed}$$ = $$\frac{s}{c}$$ = $$\frac{1.5\times10^{11}}{3\times10^8}$$ = $$500$$ seconds Therefore the it takes 500 seconds, or roughly 8 minutes and 22 seconds for the light to reach the earth from the sun. Surprisingly, this means that the sun we see in the sky is actually the sun as it was 8 minutes 22 seconds ago, hence seeing in the past.

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