To escape Earth’s gravitational field, a rocket must be launched with an initial velocity called the escape velocity.A rocket launched from the surface of Earth has velocity (in miles per second) given by v = √2GM/r + v_0^2 - 2GM/R ≈ √192000/r + v_0^2 - 48 where v_0 is the initial velocity, r is the distance from the rocket to the center of Earth, G is the gravitational constant, M is the mass of Earth, and R is the radius of Earth (approximately 4000 miles). (b) A rocket launched from the surface of the moon has velocity v (in miles per second) given by v = √1920/r + v_0^2 - 2.17. Find the escape velocity for the moon.

A rocketry club is launching model rockets. The launching pad is 30 feet above the ground. Your model rocket has an initial vertical velocity of 105 feet per second. Your friend's model rocket has an initial vertical velocity of 100 feet per second. a. Does your rocket reach a height of 200 feet? Does your friend's rocket? Explain your reasoning. b. Which rocket is in the air longer? How much longer?

At the instant of vertical launch the rocket expels exhaust at the rate of 220 kg/s with an exhaust velocity of 820 m/s. If the initial vertical acceleration is 6.80 m/$s^2$, calculate the total mass of the rocket and fuel at launch.

The third stage of a rocket fired vertically up over the north pole coasts to a maximum altitude of 500 km following burnout of its rocket motor. Calculate the downward velocity v of the rocket when it has fallen 100 km from its position of maximum altitude. (Use the mean value of $9.825 m/s^{2}$ for g and 6371 km for the mean radius of the earth.)

A ball is thrown upward with an initial velocity of $15 \mathrm{~m} / \mathrm{s}$. Using the approximate value of $g=10 \mathrm{~m} / \mathrm{s}^2$, what are the magnitude and direction of the ball's velocity:b. 2 seconds after it is thrown?

A rocket has an initial mass of $7 \times 10^{4} \mathrm{kg}$ and on firing burns its fuel at a rate of 250 kg/s. The exhaust velocity is 2500 m/s. If the rocket has a vertical ascent from resting on the earth, how long after the rocket engines fire will the rocket lift off? What is wrong with the design of this rocket?

use the position function s(t) = - 4.9t^2 + v0t + s0 for free-falling objects. A projectile is shot upward from the surface of Earth with an initial velocity of 120 meters per second. What is its velocity after 5 seconds? After IO seconds?

The concentration C of a chemical in the bloodstream t hours after injection into muscle tissue is C(t) = 3t / (27t + t³), t≥0. (a) Complete the following t: 0, 0.5, 1, 1.5, 2, 2.5, 3 and use it to approximate the time when the concentration is greatest. (b) Use a graphing utility to graph the concentration function and use the graph to approximate the time when the concentration is greatest. (c) Use calculus to determine analytically the time when the concentration is greatest.

Find the values of the constants a and b such that lim_(x→0) √(a + bx) - √3 / x = √3

Find the derivatives of the function f for n = 1, 2, 3, and 4. Use the results to write a general rule for f′(x) in terms of n.

$$f(x) = cos x / x^n$$

Find the derivatives of the function f for n = 1, 2, 3, and 4. Use the results to write a general rule for f′(x) in terms of n.

$$f(x) = x^n sin x$$

Use the graph to estimate the open intervals on which the function is increasing or decreasing. Then find the open intervals analytically. f(x) = x^4 - 2x²

Find the open intervals on which the function is increasing or decreasing.

$$y = x \sqrt{16-x^2}$$

The radius r of a circle is increasing at a rate of 3 inches per minute. Find the rates of change of the area when (a) r=6 inches and (b) b=24 inches.