Formula that predicts the location of moving enemy

Question

A rocket that explodes not on impact, but explodes on a timer will be shot from myself to the enemy. The timer can only be inputted at the start of the shot. The rocket can only go straight and has a speed of 1500m/s . The enemy is moving in a straight line at a 135 degree angle with a speed of 400m/s. The distance between myself and my enemy is 5000m. I need a formula that predicts the precise time until the rocket should explode so that it will explode directly on my enemy's location.

Answer 1

Assumptions:

 

All motion occurs in the x-y plane, and it is a horizontal plane.

The rocket launches from the origin (0,0)

Rocket moves only in a straight line at speed v = 1500 m/s

Initial position of target: (x₀,y₀).  In this problem, x₀ = 5000, y₀ = 0.

Direction target moves: θ (above +x axis).  In this problem, θ = 135°.

Speed of target: u.  In this problem, u = 400 m/s

 

You need to solve for the time-to-target t and a launch angle φ for the rocket, above the x-axis.  At time t, the rocket and target must arrive at the same point, so

 

x-coordinate of intercept point: 

 

(v cosφ)t = x₀ + (u cosθ)t

 

y-coordinate of intercept point:

 

(v sinφ)t = y₀ + (u sinθ)t

 

You have to solve a system of 2 equations in 2 unknowns t and φ:

 

(v cosφ)t = x₀ + (u cosθ)t
(v sinφ)t = y₀ + (u sinθ)t

 

This is not a linear system, so it may need to be solved by numerical methods or graphing methods.