Find the place of the explosion

[user5]

A missile is fired at speed V₀ at an angle 45⁰. The missile exploded into two parts when reaches the maximum height. The part of mass m1 is thrown back at speed V₁ relative to the missile speed before the explosion (V₁>V₀). From the shot till the explosion the wind was blown that exerts a friction force F_friction =−kV. Exactly at the explosion the wind stops to blow. Find where the explosion accrues.




Why in finding the y cordinate there is no use of the friction force? the answer says: yexplosion=V₀sin45∗t_explosion−0.5∗g∗t²_explosion while in finding the x coordinate there is a use, didn't friction have to be taken into consideration in both coordinates? Does friction force is only horizontal and not opposite the direction of the velocity?

 

[tiny-tim]

hi user5!

From the shot till the explosion the wind was blown that exerts a friction force F_friction =−kV.
…
Why in finding the y cordinate there is no use of the friction force? the answer says: yexplosion=V₀sin45∗t_explosion−0.5∗g∗t²_explosion while in finding the x coordinate there is a use, didn't friction have to be taken into consideration in both coordinates?

yes, i think you're right, and that equation is wrong

the friction seems to be opposite to the velocity,

so the governing vertical equation should be y'' + ky + g = 0

 

[user5]

thank you very much!
may you help me with solving this equation?

 

[tiny-tim]

hi user5!

there's a standard method, which is to write d/dx = D, making it:

(D² + kD + g)y = 0​

you then factor that to (D - a)(D - b)y = 0

and the solutions are:

y = C₁e^ax + C₂e^bx if a ≠ b

y = (C₁ + C₂x)e^ax if a = b
​