A) What is the minimum speed, relative to the Sun, necessary for a spacecraft to escape the solar system if it starts at the Earth's orbit?
B) Voyager I achieved a maximum speed of 125 000 km/h on its way to photograph Jupiter. Beyond what distance from the Sun is this speed sufficient to escape the solar system?
so for part A I used the equation...
V(esc)=square root of (2*G*M(earth))/(distance from the earth to the sun) is that right?
and i don't understand how to do the second part.
The escape velocity from Earth's orbit into interstellar space is 42 km/sec.
Source:
http://science.jrank.org/pages/3756/Kepler-s-Laws-Applications-generalized-forms-Kepler-s-laws.html
For you other question try:
http://en.wikipedia.org/wiki/Escape_velocity#Calculating_an_escape_velocity
If no help repost that question and a physics expert probably will get it tomorrow.
On both parts, start with the PE due to Sun
PE= INT Force*dx from inf to r, For the A, r will be Earth orbit radius, and for B), r will be the unknown.
INT GMs*M/r^2 dr= GMs*M/r
Set that equal to 1/2 GMv^2 in the first, with r= rearth orbit.
In Part A) This will change what Mass is in the equation.
In part B)
you know the KE, set that equal to GMs/r and solve for r.
For part A, you are correct in using the equation V(esc) = √(2GM(Earth)/R(Earth)) to calculate the minimum speed necessary for a spacecraft to escape the solar system from Earth's orbit. This equation calculates the escape velocity, which is the minimum velocity required for an object to escape the gravitational pull of a celestial body, in this case, the Sun.
To get the escape velocity, you need to know two things: the mass of the celestial body (in this case, the Sun) and the distance between the spacecraft and the center of the celestial body (in this case, the distance from Earth's orbit to the Sun). The value G is the gravitational constant.
In the equation, GM(Earth) is the product of the gravitational constant (G) and the mass of the Earth, R(Earth) is the distance between the Earth and the Sun. By plugging in these values into the equation, you can calculate the minimum speed required for a spacecraft to escape the solar system.
For part B, you need to determine the distance from the Sun at which the speed of 125,000 km/h achieved by Voyager 1 is sufficient to escape the solar system. To do this, you can use the concept of kinetic energy and gravitational potential energy.
The kinetic energy (KE) of an object is given by the equation KE = 1/2mv^2, where m is the mass of the object and v is its velocity. The gravitational potential energy (PE) of an object in the gravitational field of the Sun is given by the equation PE = -GMm/r, where GM is the product of the gravitational constant (G) and the mass of the Sun, m is the mass of the object, and r is the distance between the object and the center of the Sun.
To escape the solar system, the kinetic energy of the spacecraft must be equal to or greater than its gravitational potential energy. So, you can set KE = -PE and solve for r. You know the kinetic energy, which is 1/2mv^2 (where m is the mass of the spacecraft and v is its velocity), and you can substitute in the value of GM (the product of the gravitational constant and the mass of the Sun). With this equation, you can solve for the distance from the Sun at which the speed of 125,000 km/h is sufficient for the spacecraft to escape the solar system.