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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.

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.

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.

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

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