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Pavel Z.

Math and Science Tutor

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Physics (Newtonian Mechanics)

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Question:

A roller coaster cart start from rest at height h=100 m above the lowest point of the track. Find its speed at the lowest point if friction and air resistance are negligible.

Pavel Z.

Answer:

Here we can use energy conservation. At the start the cart has only gravitational potential energy $$E_{pot}=mgh$$. At the bottom the cart has no potential energy, but has some kinetic $$E_{kin}=\frac{1}{2}mv^2$$. The energy is conserved, potantial energy on the top is equal to kinetic energy at the bottom: $$mgh=\frac{1}{2}mv^2$$ => $$v^2=2gh$$ => $$v=\sqrt{2gh}=\sqrt{2 * 9.81\frac{m}{s^2} * 100 m}=44.3 \frac{m}{s}$$

Physics (Electricity and Magnetism)

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Question:

A parallel plate capacitor with capacitance C is connected to the battery that has voltage V for long time. The capacitor is disconnected from the battery and distance between its plates doubled. Find capacitance, voltage and charge on the plates after that.

Pavel Z.

Answer:

The capacitor was connected for a long time, so it is fully charged and the initial voltage across the plates is V. Charge on the plates q=C*V. After capacitor is disconnected, charge on the plates is conserved, because no current is possible. Thus, charge after the distance is doubled $$q_1=q=C*V$$. For parallel plate capacitor capacitance is given by $$C=\epsilon_0 \frac{A}{d}$$. When we double d new capacitance $$C_1=\epsilon_0 \frac{A}{2d}=\frac{1}{2}C$$. The formula $$q=C*V$$ is true always, so after we double the distance between plates $$q_1=C_1*V_1$$ or $$q=\frac{1}{2}C*V_1$$. From here $$V_1=2\frac{q}{C}=2V$$.

Calculus

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Question:

Calculate indefinite integral $$\int{sin(t)cos(t)dt}$$

Pavel Z.

Answer:

$$\int{sin(t)cos(t)dt}=\int{\frac{1}{2}sin(2t)dt}=\int{\frac{1}{4}sin(2t)d(2t)}=-\frac{1}{4}cos(2t)+C$$

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