Imagine a frictionless perfectly straight tunnel which runs from Boston (USA) to Delhi (India). We release (at zero speed) an object in the tunnel in Boston. Assume that the Earth is a perfect sphere with radius 6.4*10^3 km, with a mass of 6.0*10^24 kg and that the mass density is uniformly distributed throughout the Earth.
Question: How much time will it take the object to reach Delhi and what will be its speed when it gets to Delhi (3% accuracy is required)? You should ignore friction due to air-drag.
To find the time it takes for the object to reach Delhi and its speed at that point, we need to consider the gravitational force acting on the object as it travels through the tunnel.
1. First, let's determine the distance between Boston and Delhi. The shortest distance between two points on a sphere is given by the length of the great circle passing through those points. Considering the Earth as a perfect sphere, the distance between Boston and Delhi would be the circumference of the Earth along that latitude.
The Earth's radius is given as 6.4 * 10^3 km. The circumference of a circle is calculated as C = 2πr. So, the distance between Boston and Delhi (centered along the same latitude) is 2π * 6.4 * 10^3 km.
2. Now, let's calculate the gravitational force acting on the object as it travels through the tunnel. The force of gravity can be determined using Newton's law of gravitation, which states that F = (G * m1 * m2) / r^2, where G is the gravitational constant, m1 and m2 are the masses of the two objects, and r is the distance between their centers of mass.
In this case, the mass of the object is negligible compared to the mass of the Earth.
3. To find the acceleration experienced by the object, we can use Newton's second law of motion, F = m * a, where F is the net force and m is the mass of the object. Since the force acting on the object is the gravitational force, we can write it as mg, where g is the acceleration due to gravity.
4. The object will accelerate uniformly under the force of gravity until it reaches Delhi. We can find the time it takes for the object to reach Delhi using the equation of motion, s = ut + (1/2)at^2, where s is the distance, u is the initial velocity, a is the acceleration, and t is the time.
In this case, the initial velocity u is zero, and the distance s is the distance between Boston and Delhi.
5. Finally, to find the speed of the object when it reaches Delhi, we can use the equation v = u + at, where v is the final velocity and a is the acceleration.
By applying the above steps, we can calculate the time taken and the speed of the object accurately.