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Shashwath M.
Physics Major at the University of Texas at Dallas
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Physics (Newtonian Mechanics)
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Question:

Suppose you are an engineer who is trying to design a car. At your company, safety is important and you have been tasked with testing the car in the event of a head-on collision at $$112 \, m/s$$ in a car which has a mass of $$750 \, kg$$. How much force does the car experience if it takes 2 seconds to come to rest.

Shashwath M.

Let us first explore the relationship between force and momentum. Force can be defined as the product of mass and acceleration: $$F = ma$$. Momentum is defined as the product of mass and velocity: $$p = mv$$. The only difference between the expression for force and momentum is the second term ($$a$$ and $$v$$ respectively) . We can understand acceleration as the change of velocity over time: $$a =\frac{\Delta v}{\Delta t}$$. By substituting this in the expression for force, $$F = m\frac{\Delta v}{\Delta t}$$, and multiplying both sides by $${\Delta t}$$, we arrive at: $$F {\Delta t}= m{\Delta v}$$. $$F {\Delta t}$$ is known as impulse. Using the definitions we established, we can solve the problem by rearranging $$F {\Delta t}= m{\Delta v}$$ to solve for $${F}$$ ( $$F = \frac{m{\Delta v}}{\Delta t}$$ ). From the information provided, we know that $$m = 750 \, kg$$, $${\Delta v} = 112 \, m/s$$, and $${\Delta t} = 2\, s$$. Plugging this in to the expression for $$F$$, we get the solution: $$F = 42000 N$$.

Chemistry
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Question:

Why does the addition of one mole of $$CaCl_2$$ increase the boiling point of water more than one mole of $$NaCl$$ ?

Shashwath M.

First, we must think about why boiling point elevation occurs. When we talk about the temperature of a gas or liquid, it is the average of the kinetic energy of all the particles of the fluid. This means that even at room temperature, a glass of water can contain $$H_2O$$ molecules that have enough energy to escape into the gas phase. Now let's add some salt to the water. The salt will disassociate into $$Na^+$$ and $$Cl^-$$ ions. These ions will act as obstacles which will make it more difficult of water molecules to escape into the gas phase. This results in a higher boiling point. Now, let's compare $$NaCl$$ with $$CaCl_2$$. As we discussed earlier, $$NaCl$$ will dissociate into two ions ( $$Na^+$$ and $$Cl^-$$ ). On the other hand, $$CaCl_2$$ will dissociate into three ions ($$Ca^{2+}$$ and two $$Cl^-$$ ). This means that one mole of $$CaCl_2$$ produces more "obstacles" than one mole of $$NaCl$$, elevating the boiling point even more. The term used to describe the number of ions are produced by dissociation and amplify the effects of boiling point elevation, freezing point depression, and vapor pressure reduction is known as the Van't Hoff factor ($$i$$) and is equal to the number of ions produced by the dissociation of a compound.

Physics
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Question:

Suppose Han Solo has some difficulty accelerating his spaceship, the Millennium Falcon, to light speed. Instead, he is forced to travel at 70% the speed of light. While flying, he sees a TIE fighter approaching at a speed 80% the speed of light. Han acts quickly and fires a concussion missile at 50% the speed of light. What is the speed of the missile from the TIE fighter's point of view? ( $$c= 299792458 \, m/s$$ )

Shashwath M.

If we add the velocities of Han's ship and the missile in a Newtonian manner, the result would be 1.3 times the speed of light! Since we know that no object with mass can travel at the speed of light (or faster), we need to use another approach. This solution requires the application of relativistic velocity addition. Let's assume that $$V$$ is the speed of the Millennium Falcon, $$U$$ is the speed of the missile from Han's point of view, and $$U'$$ is the speed of the missile from the TIE fighter's point of view. Using the formula: $$U' = \frac{U+V}{1+\frac{UV}{c^2}}$$ and substituting the values from the question, we get the answer. $$U' = 0.92857c$$

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