An adult grasshopper can jump to a height of about 50 cm. i) What is the increase in the gravitational potential energy for a 0.25 g grasshopper in such a situation and ii) what is the origin of this energy?

A Cirque artist needs to swing across a river using a 15-meter rope attached to a bridge. Initially, the rope makes an angle of 60° with the vertical, and the artist swings down before rising to meet a colleague on the bank of the river. When he reaches the other bank, his rope makes a 25° angle with the vertical. Calculate the artist's speed just before he releases the rope and is caught by his colleague.

Suppose a car is traveling from point P to point R along the path PQR, and it's experiencing a force F = (12i + 6j) N. Determine the work done by the force on the car following path PQR based on the figure shown below. Can we classify this force as conservative or non-conservative?

Determine what wavelengths are present in the absorption spectrum of a hydrogen-like atom with distinct energy levels at 0.00 eV, 3.50 eV, and 5.50 eV.

The energy level diagram of a hypothetical ion is shown below. The energy levels are labeled as E₁, E₂, and E₃. Determine the wavelengths of radiation required to excite the ion from the ground state, E₁, to the 2^nd and 3^rd states.

Draw an energy level diagram for Li²⁺, showing the numerical values of n and E_n for the first four values of n.

Consider three planets, $P_1$ , $P_2$ , and $P_3$ , each with masses $m_1$ , $m_2$ , and $m_3$, initially positioned infinitely far apart. The work required to bring them into a stable equilateral triangular configuration around their common center of mass, where each side of the triangle has the same distance r, is given by:

$W=-G\left(\frac{m_1m_2}{r}+\frac{m_1m_3}{r}+\frac{m_2m_3}{r}\right)$

Does this work W correspond to the gravitational binding energy of the planetary system? Explain.