CALC ?In Section 8.6, we considered a rocket fired in outer space where there is no air resistance and where gravity is negligible. Suppose instead that the rocket is accelerating vertically upward from rest on the earth’s surface. Continue to ignore air resistance and consider only that part of the motion where the altitude of the rocket is small so that ?g ?may be assumed to be constant. (a) How is Eq. (8.37) modified by the presence of the gravity force? (b) Derive an expression for the acceleration ?a ?of the rocket, analogous to Eq. (8.39). (c) What is the acceleration of the rocket in Example 8.15 (Section 8.6) if it is near the earth’s surface rather than in outer space? You can ignore air resistance. (d) Find the speed of the rocket in Example 8.16 (Section 8.6) after 90 s if the rocket is fired from the earth’s surface rather than in outer space. You can ignore air resistance. How does your answer compare with the rocket speed calculated in Example 8.16?
Solution 112P Step 1: a) Under the influence of gravitational force, there will be an additional force “mg” which is acting towards the earth. Where “m” is the mass of the rocket and “g” is the acceleration due to gravity of the rocket. So, we can modify the equation as, dv dm mg + m dt = v ex dt This is the modified equation under the influence of gravitational force without considering the air resistance. Step 2: b) we know that, the acceleration of the rocket, a = dv/dt Then, from the equation in step 1, we can write, m dv = v dm mg dt ex dt Divide the equation throughout by “m” v dv = exdm g dt m dt Therefore, we can write, acceleration of the rocket, a = dv/dt vexdm a = ( m dt + g)