A test track for autonomous electric vehicles (EVs) features a banked curve with a certain radius, designed to test the vehicles' ability to navigate turns at a specified speed without depending on friction. This allows for testing under various environmental conditions, which can alter the coefficient of static friction between the tires and the road surface. Determine the minimum and maximum speeds, $v_{\min}$ and $v_{\max}$, at which an autonomous EV can safely handle the banked curve. These speeds should be calculated as functions of the coefficient of static friction ($\mu_{s}$), the intended test speed ($v_0$), and the curve's radius ($R$).

When a vehicle navigates a banked circular track 146 m in diameter at a uniform speed, what are the vehicle's coordinates (x and y) at this time? Given that its motion is represented in a coordinate system with its origin at the circle's center. At a given instant, the vehicle's horizontal acceleration is calculated as (-16.7 î - 24.2 ĵ) m/s².

For the following configuration of masses, determine the magnitude and direction of the sum of the gravitational force that spheres B and D exert on sphere C. Take all spheres to have the same mass equal to 6.0kg.

Consider a pair of balls (spherical) that have a collective mass of 211 kg. They are subject to a gravitational force of 6.25 x 10 ^-8 N, and their gravitational potential energy is ₋12.4 x 10 ^-8 J. Determine the individual mass of each ball in the given scenario.

A rover operating on Mars measures changes in gravitational pull as it climbs a steep hill of height ∆z, where ∆z is much smaller than R_Mars (the radius of Mars). Use the binomial expansion

(1 ± x)ⁿ = 1 ± nx + (n(n-1)/2) x² ± ...

to derive the formula approximating the change in acceleration due to gravity in Mars $\Delta g_{\text{Mars }}$, where $g_{Mars}$ is the gravitational acceleration at Mars's surface.

On a basketball court, a basketball player recorded a standing vertical leap of 125 cm. Imagine that this player was given the opportunity to travel to a spherical celestial object in outer space. The mass per unit volume of the celestial object is 2.2 g/cm³. Calculate the maximum radius of the celestial object, allowing the player to escape by jumping vertically.

An exoplanet has a mean radius equal to 0.09 times the radius of Earth and a mass equal to 0.00023 times the mass of Earth. What would be the gravitational acceleration at the exoplanet's surface?

Suppose there is a planet with a mass that is 4 times the mass of the Earth. If a spaceship lands on the planet's surface, it measures the free-fall acceleration to be 1/8 of that of the Earth's acceleration due to gravity. Determine the radius of the planet.

The Moonwatcher Explorer is a robotic spacecraft that ventures around the moon, flying low over its surface. With an orbital period of approximately 115 minutes, this spacecraft is a marvel of modern engineering. As the spacecraft glides over the rugged terrain of the moon, what free-fall acceleration does it experience on its surface?

Global positioning system (GPS) satellites use nearly round orbits about 20200 km above the earth's surface. Determine the orbital speed of a GPS satellite required to keep the satellite in orbit.

The explosion of large stars causes a supernova. The remnants of a supernova explosion may lead to the formation of a highly magnetized and compact object called a magnetar. Magnetars are a type of neutron star. Magnetars are thought to be the source of some of the most energetic and powerful explosions in the universe such as gamma-ray bursts. Magnetars rotate about their axis at a rate of once every 9.0 s. Find the radius of its geosynchronous orbit. Consider the mass of magnetar is 2.78 × 10³⁰ kg.

Scientists had sent a space probe of mass 1270 kg to an orbit around Venus. It had an orbital period of 86400 s around the planet. Given that the mass of Venus is 4.87×10²⁴ kg and the gravitational constant is G = 6.67 x 10^-11 Nm²/kg², calculate the magnitude of the gravitational force between the planet and the space probe while in orbit.