The question asks:
"An object traveling in space at velocity v0 in the x,y plane begins to feel a noticeable acceleration due to gravity from a distant planet of mass M at an angle θ between the velocity vector and the acceleration vector. Create an expression to describe the objects distance from the planet (r) at a given point in time. You may use the variables θ, a, v, r, M, and any other mathematical or physical constants to express your answer."
The basic way I'm going about this problem is setting up a manor of expressing each variable in terms of each other, but even then I find it challenging to express θ in terms of other quantities due to the nature of the acceleration at hand. I can conceptualize the math involved but I simply don't know how to put it into a an expression I can work with.
Basically I'm running with the concept that, due to the initial velocity in the x,y plane, the object's acceleration is continually increasing, resulting in a greater velocity - which in turn changes θ and thus meaning the problem is quite tricky. I initially went about this problem by setting v0 equal to zero. This results in a fairly easy integration of Newton's law of universal gravitation. (GM/r^2 = a [with an m taken from both sides]) Integrating this gives the total change in velocity over two points, then dividing by r0 gives average acceleration which you can use to determine position at a given time with some simple kinematics. But no matter how I think of the problem with a non-zero initial velocity, I can't seem to get it. I've tried using DFQs with no success. Please help, thanks.
"An object traveling in space at velocity v0 in the x,y plane begins to feel a noticeable acceleration due to gravity from a distant planet of mass M at an angle θ between the velocity vector and the acceleration vector. Create an expression to describe the objects distance from the planet (r) at a given point in time. You may use the variables θ, a, v, r, M, and any other mathematical or physical constants to express your answer."
The basic way I'm going about this problem is setting up a manor of expressing each variable in terms of each other, but even then I find it challenging to express θ in terms of other quantities due to the nature of the acceleration at hand. I can conceptualize the math involved but I simply don't know how to put it into a an expression I can work with.
Basically I'm running with the concept that, due to the initial velocity in the x,y plane, the object's acceleration is continually increasing, resulting in a greater velocity - which in turn changes θ and thus meaning the problem is quite tricky. I initially went about this problem by setting v0 equal to zero. This results in a fairly easy integration of Newton's law of universal gravitation. (GM/r^2 = a [with an m taken from both sides]) Integrating this gives the total change in velocity over two points, then dividing by r0 gives average acceleration which you can use to determine position at a given time with some simple kinematics. But no matter how I think of the problem with a non-zero initial velocity, I can't seem to get it. I've tried using DFQs with no success. Please help, thanks.