Physics Problem-Help Please?

Question

Been trying to do this problem but no luck, can ya help me, Thanks.

In this problem, you will be asked to analyze portions of a famous "thought experiment." To explain the orbital motion of the Moon, Newton imagined a cannon that could fire a cannonball horizontally at a great speed. He reasoned that the cannonball's motion was fundamentally the same as the Moon's. He linked projectile motion with orbits, bringing celestial bodies into the realm of everyday experience.

Consider the Earth to be a perfect sphere of radius r = 6.38×106 m. Imagine that the cannon is mounted on a tower 125 meters above the surface of the Earth. Ignore, as Newton did, air resistance, mountains and the motion of the Earth.

The cannon fires the cannonball horizontally, tangent to the Earth's surface. The faster the cannon fires the ball, the farther it goes before striking the ground. You wish to fire a ball at a speed so great that it never hits the ground: The spherical Earth curves down and away from it at exactly the same rate at which it falls. In this scenario the cannonball will describe a circular trajectory, circling the Earth at a constant height of 125 m. It will return to strike the cannon in the rear.

To determine the necessary speed, consider what happens during the first 1.00 km of horizontal displacement of the cannonball. You will calculate the vertical displacement, using the curvature of the Earth.

(a) Suppose you draw a line tangent to the Earth's surface at the location of the cannon tower, and extend it exactly 1.00 km in the direction of the cannonball's path. What is the angle θ (in radians) that this line segment subtends at the center of the Earth? (b) You can draw a right-angled line from the end of the red segment down to the ground in order to measure the "fall-away" y of the Earth's surface. This very nearly equals the opposite side of a triangle whose base angle is θ/2. Use tan θ/2 in an equation to approximate the Earth's vertical fall-away y in a horizontal distance of 1.00 km (state your answer as a positive number). (c) How fast must you fire the cannonball so that its fall after one horizontal kilometer equals the fall-away of the Earth? (d) Newton used the cannon as a "thought experiment" to explain (circular) orbital motion. Why is the trajectory of the cannonball not parabolic?

Answer 1

Oscar. You have written too much. This 125m tower stuff does not matter compared to r=6.38*10^6. Why is this talk about 1.00 km and angle theta? See what “zipper” above says. Besides I answer your last question << Why is the trajectory of the cannonball not parabolic?>>. Why should it be if both you and Newton demand it to be circular? Assume that V1 of cannonball is the 1st cosmic speed corresponding to a circular track. Let V2 be > than V1, then you receive an oval track. Take V3 > than V2, then the track may become parabolic, so that cannonball would never return back to the cannon to linger out among planets and stars. If you are intrigued calculate a set of velocities and corresponding orbits of the cannonball. If cannon is not available consider a football! I like your inventive wits – to be honest!

Answer 2

you have to use the concept of centripetal force. Set the force equal to gravity 9.8 m/sec^2 or 32.2 ft/ sec^2 and sovle for velocity.