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Tonikaku Kawaii. A projectile of mass m is launched with initial velocity o at angle e from the +x direction. From the origin, the projectile is connected to a massless spring with force constant k, as shown. Eet-up the equations of motion of the projectile, both in the horizontal and vertical directions. A.

Tonikaku Kawaii. A projectile of mass m is launched with initial velocity o at angle e from the +x direction. From the origin, the projectile is connected to a massless spring with force constant k, as shown. Eet-up the equations of motion of the projectile, both in the horizontal and vertical directions. A.

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Hi! I already have answers for parts A-C in this problem, but I'm having trouble with the last part (D.). Attached is the guide to answer it. 

My answers for part C are:

A = 0

B = (V_o cos theta)/omega

C = -(mg)/k

D = (V_o sin theta)/omega

E = mg/k

I tried substituting these coefficients to the x(t) and y(t) formulas, but I still can't reduce them to the usual projectile kinematic equations. Also, what does it mean to do a small angle approximation?

 

I hope you can help, thank you in advanced!

6. Tonikaku Kawaii. A projectile of mass m is launched with initial velocity vo at angle e from the +x direction. From the origin, the projectile is connected to a massless spring with force constant k, as shown. A. Set-up the equations of motion of the projectile, both in the horizontal and vertical directions. В. Observe from the equations of motion that the projectile is simple harmonic in the x direction, but not necessarily in the y direction. However, if we let : = y + a, where a is a constant, one can show that we can convert the equation of motion in y to a form given by k which is now simple harmonic in z. What must be the value of a for this to be true? C. From the equations of motion, one can show that the horizontal and vertical positions of the mass as functions of time can be written as x(t) = A cos wt +B sin wt y(t) = C cos wt + D sin wt + E provided that the coefficients satisfy the conditions that initially, x(0) = y(0) = 0, v,(t = 0) = vg cos 8, and v, (t = 0) = vo sin 8. Determine the coefficients A, B, C, D, and E in terms of the given variables.
Transcribed Image Text:6. Tonikaku Kawaii. A projectile of mass m is launched with initial velocity vo at angle e from the +x direction. From the origin, the projectile is connected to a massless spring with force constant k, as shown. A. Set-up the equations of motion of the projectile, both in the horizontal and vertical directions. В. Observe from the equations of motion that the projectile is simple harmonic in the x direction, but not necessarily in the y direction. However, if we let : = y + a, where a is a constant, one can show that we can convert the equation of motion in y to a form given by k which is now simple harmonic in z. What must be the value of a for this to be true? C. From the equations of motion, one can show that the horizontal and vertical positions of the mass as functions of time can be written as x(t) = A cos wt +B sin wt y(t) = C cos wt + D sin wt + E provided that the coefficients satisfy the conditions that initially, x(0) = y(0) = 0, v,(t = 0) = vg cos 8, and v, (t = 0) = vo sin 8. Determine the coefficients A, B, C, D, and E in terms of the given variables.
D. show that x(t) and y(t) reduce to usual projectile kinematic equations for w very small. Guide starts here: You must do a small angle approximation for this exercise. sin 8 xe for 8 «1 Why is this true? You can express the sine function as an infinite series: (-1)* 2k+1 (2k + 1)! sin x = =x- 3! 5! When e «1, the higher order terms are very small. Hence, we can approximate sin e as 0. What about the cosine function? sin? e + cos? 8 = 1 cos? 8 = 1- sin? e - cos e = V1- sin? 6 When e « 1, cos e = /1- sin? 0 1-02 = 1 Hence, cos e x 1 for 8 « 1. Use these to solve for x(t) and y(t) in the small w regime.
Transcribed Image Text:D. show that x(t) and y(t) reduce to usual projectile kinematic equations for w very small. Guide starts here: You must do a small angle approximation for this exercise. sin 8 xe for 8 «1 Why is this true? You can express the sine function as an infinite series: (-1)* 2k+1 (2k + 1)! sin x = =x- 3! 5! When e «1, the higher order terms are very small. Hence, we can approximate sin e as 0. What about the cosine function? sin? e + cos? 8 = 1 cos? 8 = 1- sin? e - cos e = V1- sin? 6 When e « 1, cos e = /1- sin? 0 1-02 = 1 Hence, cos e x 1 for 8 « 1. Use these to solve for x(t) and y(t) in the small w regime.
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