Projectile Problem: Find Acceleration of Missile to Shoot Down ICBM

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The discussion centers on calculating the acceleration of a missile intended to intercept an ICBM launched at a 70-degree angle with an initial speed of 5100 m/s. The missile is launched 20 seconds after the ICBM and must reach a height of 1000 km. Participants confirm the logic of using kinematic equations to find the time and distance traveled by both projectiles, noting that the ICBM's trajectory allows it to reach the 1000 km height twice. Corrections are suggested regarding the use of initial velocity in the Xf equation, emphasizing that the missile's motion is one-dimensional. Overall, the approach to solving the problem is validated, with minor adjustments recommended for accuracy.
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Homework Statement



An ICBM is launched from the surface with an initial angle of 70 degrees and initial speed of 5100 [m/s]. 500 [km] behind the launch site a cruise missile is launched to shoot it down at the height of 1000 [km]. Find the acceleration of the missile if it is launched 20 after the ICBM.

Note: the ICBM is a free fall projectile but the missile moves a long a straight line.

Homework Equations



xf = vi*t + (1/2)at^2/
yf = vi*sin(theta)*t - 4.9t^2

The Attempt at a Solution



I just want to make sure my logic is correct on this one-

Alright, I know there will be two answers seeing as the ICBM passes y = 1000 [km] twice. And I also know that the missile is going to be considered 1D motion. So, what I did was plug in all of the knowns into the Yf(where Yf = 1000) and get two values for time.

Here is what I did next...

I find the ICBM's DeltaX by using the known angle and side of the ICBM. I then added that DeltaX to the 500km that the missile was behind it. Using Pythagorean theorem(with that answer and the given 1000km), I got the total distance the missile traveled. I plugged into my first equation and with the time(adjusted for the late takeoff) and solved for acceleration. I did the same thing for the second time/location.


I got something like 33.3 [m/s] for the first one... but I'm more interested if my logic is correct.

Thanks!
 
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Your logic looks good.
The first thing that I'm aware is your Xf equation. the initial velocity in x direction should have v_i cos(\theta)
And an acceleration in that direction is zero.
 
The Xf equation is for the missile which, from how I interpreted it, 1D movement... i.e., along a straight line.
 
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