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
​

 

[paranormal]

I would first clarify that the given information is not enough to accurately determine the exact location of the explosion. However, based on the given information, we can make some assumptions and provide a response.

Firstly, it is important to note that the friction force in this scenario is only acting horizontally and not in the opposite direction of the velocity. This is because the wind is blowing in the same direction as the velocity of the missile, therefore, the friction force will only act in the direction opposite to the wind's direction.

In terms of finding the y-coordinate of the explosion, we do not need to take into consideration the friction force because it is acting horizontally and does not affect the vertical motion of the missile. Therefore, the equation for the y-coordinate only includes the initial velocity, the angle of launch, and the acceleration due to gravity.

On the other hand, in finding the x-coordinate, we do need to take into consideration the friction force because it affects the horizontal motion of the missile. The equation for the x-coordinate includes the initial velocity, the angle of launch, the acceleration due to gravity, and the friction force.

In conclusion, the friction force is only acting horizontally and does not affect the vertical motion of the missile. Therefore, it is not included in the equation for the y-coordinate of the explosion. However, it does affect the horizontal motion, and hence, it is included in the equation for the x-coordinate.