an aeroplane of mass 8000kg flying at 600km.h-1 fires a projectile of mass 100kg at a speed of 900 km.h-1. Calculate the velocity of the aeroplane after firing if the projectile was fired a)forward b) backward

To calculate the final velocity of the airplane after firing a projectile, we need to consider the principles of conservation of momentum.

The momentum is given by the product of mass and velocity, p = m * v.

Let's denote the initial velocity of the airplane as V_Airplane_initial, the mass of the airplane as M_Airplane, the velocity of the projectile as V_Projectile, and the mass of the projectile as M_Projectile.

The initial momentum of the airplane before firing is given by:

P_Airplane_initial = M_Airplane * V_Airplane_initial

The initial momentum of the projectile before being fired is given by:

P_Projectile_initial = M_Projectile * 0 (since the projectile is stationary before being fired)

After firing the projectile, the total momentum is conserved, so the final momentum of the system is:

P_Final = (M_Airplane * V_Airplane_final) + (M_Projectile * V_Projectile_final)

Now let's calculate the final velocity of the airplane after firing the projectile:

a) If the projectile is fired forward:

The final velocity of the projectile relative to the airplane (V_Projectile_relative) would be the difference between the velocity of the projectile and the velocity of the airplane:

V_Projectile_relative = V_Projectile - V_Airplane_initial

Since the projectile is fired forward, the final velocity of the projectile relative to the airplane is positive (V_Projectile_relative > 0). Therefore, the final velocity of the airplane after firing is:

M_Airplane * V_Airplane_final = M_Airplane * V_Airplane_initial + M_Projectile * V_Projectile_relative

V_Airplane_final = (M_Airplane * V_Airplane_initial + M_Projectile * V_Projectile_relative) / M_Airplane

Now let's substitute the given values into the equation and calculate V_Airplane_final:

M_Airplane = 8000 kg
V_Airplane_initial = 600 km/h = 600 * (1000 m/3600 s) = 166.67 m/s
M_Projectile = 100 kg
V_Projectile = 900 km/h = 900 * (1000 m/3600 s) = 250 m/s

V_Projectile_relative = V_Projectile - V_Airplane_initial = 250 m/s - 166.67 m/s = 83.33 m/s

V_Airplane_final = (8000 kg * 166.67 m/s + 100 kg * 83.33 m/s) / 8000 kg

Simplifying the equation, we get:

V_Airplane_final ≈ 166.67 m/s + 0.10416 m/s ≈ 166.77 m/s

Therefore, the velocity of the airplane after firing the projectile forward is approximately 166.77 m/s.

b) If the projectile is fired backward:

The final velocity of the projectile relative to the airplane (V_Projectile_relative) would be the sum of the velocity of the projectile and the velocity of the airplane:

V_Projectile_relative = V_Projectile + V_Airplane_initial

Since the projectile is fired backward, the final velocity of the projectile relative to the airplane is negative (V_Projectile_relative < 0). Therefore, the final velocity of the airplane after firing is:

M_Airplane * V_Airplane_final = M_Airplane * V_Airplane_initial + M_Projectile * V_Projectile_relative

V_Airplane_final = (M_Airplane * V_Airplane_initial + M_Projectile * V_Projectile_relative) / M_Airplane

Now let's substitute the given values into the equation and calculate V_Airplane_final:

M_Airplane = 8000 kg
V_Airplane_initial = 600 km/h = 600 * (1000 m/3600 s) = 166.67 m/s
M_Projectile = 100 kg
V_Projectile = 900 km/h = 900 * (1000 m/3600 s) = 250 m/s

V_Projectile_relative = V_Projectile + V_Airplane_initial = 250 m/s + 166.67 m/s = 416.67 m/s

V_Airplane_final = (8000 kg * 166.67 m/s + 100 kg * 416.67 m/s) / 8000 kg

Simplifying the equation, we get:

V_Airplane_final ≈ 166.67 m/s + 0.52 m/s ≈ 167.19 m/s

Therefore, the velocity of the airplane after firing the projectile backward is approximately 167.19 m/s.