Question

Suppose that on earth you can jump straight up a distance of 36 cm. Asteroids are made of material with mass density 2800 kg/m^3. What is the maximum diameter of a spherical asteroid from which you could escape by jumping? Express your answer with the appropriate units

Answer 1

Diameter, d = 4237.14 meters

Step-by-step explanation:

It is given that,

On earth you can jump straight up a distance of 36 cm, h = 36 cm = 0.36 m

Asteroids are made of material with mass density,
`d=2800\ kg/m^3`

Escape velocity is given by :

`v=\sqrt{(2GM)/(R)}`

`R=(2GM)/(v^2)`

Where

G is the universal gravitational constant

M is the mass

Density,
`d=(M)/(V)`

Density,
`d=(M)/(4/3 \pi R^3)`

`M=2800* (4)/(3)\pi * R^3=11728.61\ R^3`..............(2)

Now using conservation of energy as :

`(1)/(2)mv^2=mgh`

`v=√(2gh) =√(2* 9.8* 0.36)=2.65\ m/s`

`R=(2* 6.67* 10^(-11)* 11728.61\ R^3)/((2.65)^2)`

`R^2=((2.65)^2)/(2* 6.67* 10^(-11)* 11728.61)`

Radius, R = 2118.57 m

Diameter, d = 2R

`d=2* 2118.57=4237.14\ m`

So, the maximum diameter of a spherical asteroid from which you could escape by jumping is 4237.14 meters. Hence, this is the required solution.