Answer:
5120 m/s
Explanation:
The acceleration due to gravity is:
g = MG / r²
where M is the mass of the earth, G is the universal constant of gravitation, and r is the distance from the earth's center to the object's center.
Here, r = h + R, where h is the height of the chest above the surface and R is the radius of the earth.
g = MG / (h + R)²
Acceleration is the derivative of velocity:
dv/dt = MG / (h + R)²
Using chain rule, we can say:
(dv/dh) (dh/dt) = MG / (h + R)²
(dv/dh) v = MG / (h + R)²
Separate the variables:
v dv = MG / (h + R)² dh
Integrating:
∫₀ᵛ v dv = MG ∫₀ʰ dh / (h + R)²
½ v² |₀ᵛ = -MG / (h + R) |₀ʰ
½ (v² − 0²) = -MG / (h + R) − -MG / (0 + R)
½ v² = -MG / (h + R) + MG / R
½ v² = MGh / (R(h + R))
v² = 2MGh / (R(h + R))
Given:
M = 5.98×10²⁴ kg
R = 6.37×10⁶ m
h = 1.69×10⁶ m
G = 6.67×10⁻¹¹ m³/kg/s²
Plugging in:
v² = 2 (5.98×10²⁴) (6.67×10⁻¹¹) (1.69×10⁶) / ((6.37×10⁶) (1.69×10⁶ + 6.37×10⁶))
v² = 2 (5.98) (6.67) (1.69) / ((6.37) (1.69 + 6.37)) × 10⁷
v ≈ 5120 m/s
Notice that if we had approximated g as a constant 9.8 m/s², we would have gotten an answer of:
v² = v₀² + 2a(x - x₀)
v² = (0 m/s)² + 2 (9.8 m/s²) (1.69×10⁶ m - 0 m)
v ≈ 5760 m/s
So we know that our calculated velocity of 5120 m/s is a reasonable answer.
