A fighter-jet is in level flight at an altitude of 400m and with a speed of 200m/s. The fighter launches a rocket-powered missile horizontally from the underside of the jet. The rocket engine provided a constant horizontal acceleration of 25.0 m/s^2.

a) how far does the missile travel horizontally from the launch point to impact with the ground?

b) calculate the components of the missile's impact velocity.

x: v(x) = v₀ +at,

x= v₀•t+at²/2,
y: v(y) = gt,
h=gt²/2,
t=sqrt(2h/g) = sqrt(2•400/9.8)= 9.04 s.
v(y) = gt = 9.8•9.04 =88.6 m/s,
v(x) = v₀ +at = 200+25•9.04=426 m/s.
s= 200•9.04 + 25•9.04²/2 = 2829.5 m.

To solve this problem, we can break it down into two parts: horizontal motion and vertical motion. Let's begin!

a) To determine the horizontal distance traveled by the missile, we need to find the time it takes for the missile to hit the ground. We can use the vertical motion equation:

h = ut + (1/2)at^2

In this equation:
- h represents the initial vertical displacement (the altitude of the fighter-jet), which is 400m.
- u represents the initial vertical velocity of the missile, which is 0 m/s since it is launched horizontally.
- a represents the vertical acceleration experienced by the missile due to gravity, which is -9.8 m/s^2 (assuming downward is positive).
- t is the time it takes for the missile to fall and hit the ground.

By substituting the given values into the equation, we get:

400 = 0*t + (1/2)(-9.8)*(t^2)

Simplifying the equation, we have:

4.9t^2 = 400

Dividing both sides by 4.9, we get:

t^2 = 81.632

Taking the square root of both sides, we find:

t ≈ 9.043 seconds

Now that we have the time taken for the missile to hit the ground, we can find the horizontal distance traveled using the horizontal motion equation:

s = ut + (1/2)at^2

In this equation:
- s represents the horizontal displacement or the distance traveled by the missile.
- u represents the initial horizontal velocity of the missile, which is the same as the velocity of the fighter-jet since it is launched horizontally. Therefore, u = 200 m/s.
- a represents the horizontal acceleration experienced by the missile, which is 25 m/s^2.
- t is the time taken for the missile to hit the ground, which we found to be 9.043 seconds.

Substituting the values into the equation, we have:

s = 200*9.043 + (1/2)*25*(9.043^2)

Evaluating the equation, we get:

s ≈ 1810.56 meters

Therefore, the missile travels approximately 1810.56 meters horizontally from the launch point to impact with the ground.

b) To calculate the components of the missile's impact velocity, we can use the following formulas:

Vertical component:
v_vertical = u + at

In this equation:
- v_vertical represents the vertical component of the missile's impact velocity.
- u represents the initial vertical velocity, which is 0 m/s.
- a represents the vertical acceleration experienced by the missile due to gravity, which is -9.8 m/s^2 (assuming downward is positive).
- t is the time taken for the missile to hit the ground, which we found to be 9.043 seconds.

Substituting the values, we have:

v_vertical = 0 + (-9.8)*9.043

Evaluating the equation, we get:

v_vertical ≈ -88.64 m/s

Therefore, the vertical component of the missile's impact velocity is approximately -88.64 m/s, indicating a downward direction.

For the horizontal component, we know that it remains constant throughout the motion since there is no horizontal acceleration. Therefore, the horizontal component of the missile's impact velocity is equal to its initial horizontal velocity, which is 200 m/s.

Hence, the components of the missile's impact velocity are approximately:
- Vertical component: -88.64 m/s (downward)
- Horizontal component: 200 m/s

Note: The negative sign for the vertical component indicates a downward direction, following the commonly used convention.