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baron.cecil
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I first want to say that this isn't a problem from school or anything, I just thought of it one day and when I tried to do it, I couldn't!
If the Earth suddenly stopped orbiting the sun in its circular path, it would immediately begin the accelerate toward the sun in a straight path. From a classical kinematic point of view, how long will it take the Earth to reach the sun if r(0)=ri (distance from Earth to sun), v(0)=0, and a(0)=0.
I understand classical kinematics (a=dv/dt=d^2x/t^2), but in a macroscopic case like this, acceleration isn't constant; its a function of position, according to Newtons Law of gravitation a=G*m/r(t)^2.
Newton's law of gravitation: A smaller object will accelerate towards a larger object with an acceleration = G*m/r(t)^2, where G is the gravitational constant, m is the mass of the bigger object, r(t) is the distance between the two objects.
The first thing I thought to do was integrate a=G*m/r(t)^2 twice with time to get s as a function of t. => v=G*m*t/r^2 => s=G*m*t^2/(2*r^2) and s(ti)=r and s(tf)=0. I don't know where to go from there because of I have position as a function of time and position (if that makes sense?)
So r(t)=ri - s. => s=ri - r(t) => ri - r(t) = G*m*t^2/(2*r^2).
Can anyone help me out with this one? Thanks!
Homework Statement
If the Earth suddenly stopped orbiting the sun in its circular path, it would immediately begin the accelerate toward the sun in a straight path. From a classical kinematic point of view, how long will it take the Earth to reach the sun if r(0)=ri (distance from Earth to sun), v(0)=0, and a(0)=0.
I understand classical kinematics (a=dv/dt=d^2x/t^2), but in a macroscopic case like this, acceleration isn't constant; its a function of position, according to Newtons Law of gravitation a=G*m/r(t)^2.
Homework Equations
Newton's law of gravitation: A smaller object will accelerate towards a larger object with an acceleration = G*m/r(t)^2, where G is the gravitational constant, m is the mass of the bigger object, r(t) is the distance between the two objects.
The Attempt at a Solution
The first thing I thought to do was integrate a=G*m/r(t)^2 twice with time to get s as a function of t. => v=G*m*t/r^2 => s=G*m*t^2/(2*r^2) and s(ti)=r and s(tf)=0. I don't know where to go from there because of I have position as a function of time and position (if that makes sense?)
So r(t)=ri - s. => s=ri - r(t) => ri - r(t) = G*m*t^2/(2*r^2).
Can anyone help me out with this one? Thanks!