Calculate the escape velocity from Earth using the formula v_e = sqrt(2gR) with g=9.8 m/s2 and R=6.4 × 106 m.

Question

ScanSolve AI Tutor · Accepted Answer

To calculate the escape velocity, we use the formula:

v_e = \sqrt{2gR}

where:

$v_e$ is the escape velocity
$g$ is the acceleration due to gravity on the Earth's surface ( $9.8 \text{ m/s}^2$ )
$R$ is the radius of the Earth ( $6.4 \times 10^6 \text{ m}$ )

Substitute the given values into the formula.
$v_e = \sqrt{2 \times 9.8 \text{ m/s}^2 \times 6.4 \times 10^6 \text{ m}}$
Perform the multiplication inside the square root.
$v_e = \sqrt{19.6 \text{ m/s}^2 \times 6.4 \times 10^6 \text{ m}}$ $v_e = \sqrt{125.44 \times 10^6 \text{ m}^2/\text{s}^2}$
Calculate the square root.
$v_e = \sqrt{125.44} \times \sqrt{10^6} \text{ m/s}$ $v_e \approx 11.20 \times 10^3 \text{ m/s}$ $v_e \approx 11200 \text{ m/s}$
Convert the velocity from meters per second (m/s) to kilometers per second (km/s).
$v_e = 11200 \text{ m/s} \times \frac{1 \text{ km}}{1000 \text{ m}}$ $v_e \approx 11.2 \text{ km/s}$
Comparing this value to the given options:
A. $25.3 \text{ km/s}$
B. $4.2 \text{ km/s}$
C. $4.0 \text{ km/s}$
D. $11.3 \text{ km/s}$

The calculated escape velocity of $11.2 \text{ km/s}$ is closest to option D.

The final answer is $\boxed{\text{D. } 11.3 \text{ km/s}}$ .

Calculate the escape velocity from Earth using the formula v_e = sqrt(2gR) with g=9.8 m/s2 and R=6.4 x 106 m.