**Problem Statement:**A spaceship's orbital maneuver requires a speed increase of \( 1.00 \times 10^3 \) m/s. If its engine has an exhaust speed of \( 2.90 \times 10^3 \) m/s, determine the required ratio \( \frac{M_i}{M_f} \) of its initial mass to its final mass. (The difference \( M_i - M_f \) equals the mass of the ejected fuel.)**Explanation and Solution:**To solve this problem, we need to use the Tsiolkovsky rocket equation (also known as the ideal rocket equation), which relates the initial and final masses of a rocket to the velocity change and exhaust velocity. The equation is:\[\Delta v = v_e \ln \left( \frac{M_i}{M_f} \right)\]Where:- \(\Delta v\) is the change in velocity (speed increase) required.- \(v_e\) is the exhaust velocity.- \(M_i\) is the initial mass of the spaceship.- \(M_f\) is the final mass of the spaceship.- \(\ln\) denotes the natural logarithm.Given data:- \(\Delta v = 1.00 \times 10^3 \text{ m/s}\)- \(v_e = 2.90 \times 10^3 \text{ m/s}\)Step 1: Rearrange the Tsiolkovsky rocket equation to solve for \( \frac{M_i}{M_f} \):\[\frac{M_i}{M_f} = e^{\frac{\Delta v}{v_e}}\]Step 2: Substitute the given values into the equation:\[\frac{M_i}{M_f} = e^{\frac{1.00 \times 10^3 \text{ m/s}}{2.90 \times 10^3 \text{ m/s}}}\]Step 3: Calculate the exponent:\[\frac{1.00 \times 10^3}{2.90 \times 10^3} = \frac{1}{2.9} \approx 0.3448\]Step 4: Use the exponential function:\[\frac{M_i}{M_f} = e^{0.3448} \approx

Given information: Here, Δv is the increase in speed, and ve is the exhaust speed.

A spaceship's orbital maneuver requires a speed increase of 1.00 x 103 m/s. If its engine has an exhaust speed of 2.90 × 103 m/s, determine the required ratio ". of its initial mass to its final mass. (The difference m, - M, Mf equals the mass of the ejected fuel.)

A spaceship's orbital maneuver requires a speed increase of 1.00 x 103 m/s. If its engine has an exhaust speed of 2.90 × 103 m/s, determine the required ratio ". of its initial mass to its final mass. (The difference m, - M, Mf equals the mass of the ejected fuel.)

College Physics

11th Edition

ISBN:9781305952300

Author:Raymond A. Serway, Chris Vuille

Publisher:Raymond A. Serway, Chris Vuille

Chapter1: Units, Trigonometry. And Vectors

Section: Chapter Questions

Problem 1CQ: Estimate the order of magnitude of the length, in meters, of each of the following; (a) a mouse, (b)...

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