We will use differential equations to model the orbits and locations of Earth, Mars, and the spacecraft using Newton’s two laws mentioned above. Newton’s second law of motion in vector form is: F^→=ma^→ (1) where F^→ is the force vector in N (Newtons), and a^→ is the acceleration vector in m/s^2,and m is the mass in kg. Newton’s law of gravitation in vector form is: F^→=GMm/lr^→l*r^→/lr^→l where G=6.67x10^-11 m^3/s^2*kg is the universal gravitational constant, M is the mass of the larger object (the Sun), and is 2x10^30 kg, and m is the mass the smaller one (the planets or the spacecraft). The vector r^→ is the vector connecting the Sun to the orbiting objects. Step one ) The motion force in Equation(1), and the gravitational force in Equation(2) are equal. Equate the right hand sides of equations (1) and (2), and cancel the common factor on the left and right sides. Answer: f^→=ma^→ f=Gmm/lr^→l^2 a^→=Gmm/lr^→l^2 x r^→/lr^→l r^→=r^→/lr^→l * Gmm Could you please help me in step 2 and 2.1 ? 2) Using ) a^→(t) = r^→(t ) , and r ^→(t )=x(t i^→+y(t) j ^→ convert the equation in step 1 above to an equation involving x(t ), y(t) and their second derivatives. Note that these are the xy- coordinates of each orbiting object. 2.1) When two vectors are equal, their components are equal (this means that when ai^→+bj^→ =ci^→+ji^→, then a = c and b = d ). Equate the x and the y components on each side of the equation you got in step 2. This should give you two second-order differential equations, one involving x′′(t) , x(t) , and y(t), and the other involving y′′(t) , x(t), and y(t) . These equations will not have i^→ and j^→ in them.
We will use differential equations to model the orbits and locations of Earth, Mars, and the spacecraft using Newton’s two laws mentioned above. Newton’s second law of motion in vector form is: F^→=ma^→ (1) where F^→ is the force vector in N (Newtons), and a^→ is the acceleration vector in m/s^2,and m is the mass in kg. Newton’s law of gravitation in vector form is: F^→=GMm/lr^→l*r^→/lr^→l
where G=6.67x10^-11 m^3/s^2*kg is the universal gravitational constant, M is the mass of the larger object (the Sun), and is 2x10^30 kg, and m is the mass the smaller one (the planets or the spacecraft). The vector r^→ is the vector connecting the Sun to the orbiting objects.
Step one ) The motion force in Equation(1), and the gravitational force in Equation(2) are equal. Equate the right hand sides of equations (1) and (2), and cancel the common factor on the left and right sides.
Answer: f^→=ma^→
f=Gmm/lr^→l^2
a^→=Gmm/lr^→l^2 x r^→/lr^→l
r^→=r^→/lr^→l * Gmm
Could you please help me in step 2 and 2.1 ?
2) Using ) a^→(t) = r^→(t ) , and r ^→(t )=x(t i^→+y(t) j ^→ convert the equation in step 1 above to an equation involving x(t ), y(t) and their second derivatives. Note that these are the xy- coordinates of each orbiting object.
2.1) When two vectors are equal, their components are equal (this means that when ai^→+bj^→ =ci^→+ji^→, then a = c and b = d ). Equate the x and the y components on each side of the equation you got in step 2. This should give you two second-order differential equations, one involving x′′(t) , x(t) , and y(t), and the other involving y′′(t) , x(t), and y(t) . These equations will not have i^→ and j^→ in them.
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