The escape speed from a planet of mass 2.5×10^24 is 7.0 km/s.What is the planet's radius?
Iam using the formula v=sqr(2GM/r)
v=7
G=6.67*10^-11
M=2.5*10^24
r=?
I get 6.8*10^12
Where did I go wrong?
To solve for the radius (r) using the escape speed formula v = sqrt(2GM/r), you correctly identified the values for v (escape speed), G (gravitational constant), and M (mass of the planet). However, it seems that you made a calculation mistake while solving for r.
Let's go step by step to determine the correct solution:
Given:
v = 7.0 km/s
G = 6.67 x 10^(-11) m^3/(kg s^2)
M = 2.5 x 10^(24) kg
r = ?
The formula can be rearranged to solve for r:
v = sqrt(2GM/r)
Squaring both sides of the equation, we get:
v^2 = 2GM/r
Now, let's substitute the known values into the equation:
(7.0 km/s)^2 = 2 * (6.67 x 10^(-11) m^3/(kg s^2)) * (2.5 x 10^(24) kg) / r
Before we continue, let's convert the escape speed (v) from km/s to m/s by multiplying it by 1000:
v = 7.0 km/s * 1000 = 7000 m/s
(7000 m/s)^2 = 2 * (6.67 x 10^(-11) m^3/(kg s^2)) * (2.5 x 10^(24) kg) / r
Simplifying the equation:
49,000,000 m^2/s^2 = 33.35 x 10^(13) m^3/(kg s^2) * 2.5 x 10^(24) kg / r
Cancelling units and simplifying further:
49,000,000 = 83.375 x 10^(13) * 10^(24) / r
49,000,000 = 208.4375 x 10^(37) / r
Multiplying both sides by r:
49,000,000r = 208.4375 x 10^(37)
Dividing both sides by 49,000,000:
r = (208.4375 x 10^(37)) / 49,000,000
Calculating this result, we get:
r ≈ 4.25 x 10^(26) meters
So, the correct radius of the planet is approximately 4.25 x 10^(26) meters.