Question

In Example 2.6, we considered a simple model for a rocket launched from the surface of the Earth. A better expression for a rocket's position measured from the center of the Earth is given by y(t) = RE3/2 + 3 g 2 REt 2/3 where RE is the radius of the Earth (6.38 ✕ 106 m) and g is the constant acceleration of an object in free fall near the Earth's surface (9.81 m/s2). (a) Derive expressions for vy(t) and ay(t). (Use the following as necessary: g, RE, and t. Do not substitute numerical values; use variables only.) vy(t) = √ g 2 ​2R E ​(R ( 3 2 ​) E ​+3√ g 2 ​R E ​t)(− 1 3 ​) m/s ay(t) = m/s2 (b) Plot y(t), vy(t), and ay(t). (A spreadsheet program would be helpful. Submit a file with a maximum size of 1 MB.) This answer has not been graded yet. (c) When will the rocket be at y = 4RE? Your response differs significantly from the correct answer. Rework your solution from the beginning and check each step carefully. s (d) What are vy and ay when y = 4RE? (Express your answers in vector form.) vy(t) = m/s ay(t) = m/s2

Answer 1

A better expression for a rocket’s position which measured from the center of Earth is given by -

y (t) = [R_E^3/2 + 3 (g / 2) R_E t]^2/3

where, R_E = radius of the Earth = 6.38 x 10⁶ m

g = constant acceleration of an object in free fall near the Earth's surface = 9.81 m/s²

(a) Derive an expressions for velocity, v_y(t) and acceleration, a_y(t) of the rocket.

we know that, velocity of the rocket is given by -

v_y (t) = d y(t) / dt

v_y (t) = d {R_E^3/2 + 3 (g / 2) R_E t]^2/3} / dt

v_y (t) = (2/3) [3 R_E(g / 2)] / {R_E^3/2 + 3 (g / 2) R_E t}^1/3

v_y (t) = (R_E2g) / {R_E^3/2 + 3 (g / 2) R_E t}^1/3

we know that, acceleration of the rocket is given by -

a_y (t) = d² y(t) / dt

a_y (t) = d² {R_E^3/2 + 3 (g / 2) R_E t]^2/3} / dt

a_y (t) = - (2/9) [9 R_E² (g / 2)] / {R_E^3/2 + 3 (g / 2) R_E t}^4/3

a_y (t) = - (g R_E²) / {R_E^3/2 + 3 (g / 2) R_E t}^4/3

(c) When will the rocket be at y = 4 R_E?

y = [R_E^3/2 + 3 (g / 2) R_E t]^2/3

4 R_E = [R_E^3/2 + 3 (g / 2) R_E t]^2/3

64 R_E³ = R_E^3/2 + 3 (g / 2) R_E t

[3 (g / 2)] t = 7 R_E

t = (7 / 3) (2 R_E / g)

(d) What are v_y and a_y when y = 4 R_E?

we know that, v_y = (R_E2g) / {R_E^3/2 + 3 (g / 2) R_E t}^1/3

v_y = (R_E2g) / {R_E^3/2 + 3 (g / 2) R_E [(7 / 3) (2 R_E / g)]}^1/3

v_y = (R_E2g) / (8 R_E^3/2)^1/3

v_y = g R_E / 2

v_y = [(9.81 m/s²) (6.38 x 10⁶ m)] / 2

v_y = 31293900 m²/s²

v_y = 5594.1 m/s

we know that, a_y = - (g R_E²) / {R_E^3/2 + 3 (g / 2) R_E t}^4/3

a_y = - (g R_E²) / {R_E^3/2 + 3 (g / 2) R_E [(7 / 3) (2 R_E / g)]}^4/3

a_y = - (g R_E²) / (8 R_E^3/2)^4/3

a_y = - (g R_E²) / (16 R_E²)

a_y = - g / 16 - [(9.81 m/s²) / (16)]

a_y = - 0.613 m/s²