A missile of mass 1.20 102 kg is fired from a plane of mass 5.50 103 kg initially moving at a speed of 3.05 102 m/s. If the speed of the missile relative to the plane is 1.21 103 m/s, what is the final velocity of the plane?
To solve this problem, we can apply the law of conservation of momentum. According to this law, the total momentum before the firing of the missile is equal to the total momentum after the firing.
The momentum of an object is given by the product of its mass and velocity.
Let's assume the final velocity of the plane is V_f.
The initial momentum of the system (plane + missile) is the sum of the momentum of the plane and the missile:
P_initial = (mass of plane) * (initial velocity of plane) + (mass of missile) * (initial velocity of missile)
P_initial = (5.50 * 10^3 kg) * (3.05 * 10^2 m/s) + (1.20 * 10^2 kg) * (1.21 * 10^3 m/s)
The final momentum of the system is the sum of the momentum of the plane and the momentum of the missile relative to the plane:
P_final = (mass of plane) * (final velocity of plane) + (mass of missile) * (relative velocity of missile to plane)
P_final = (5.50 * 10^3 kg) * (V_f) + (1.20 * 10^2 kg) * (1.21 * 10^3 m/s - V_f)
According to the law of conservation of momentum, P_initial = P_final.
Therefore, we can set the two equations equal to each other and solve for V_f:
(5.50 * 10^3 kg) * (3.05 * 10^2 m/s) + (1.20 * 10^2 kg) * (1.21 * 10^3 m/s) = (5.50 * 10^3 kg) * (V_f) + (1.20 * 10^2 kg) * (1.21 * 10^3 m/s - V_f)
By solving this equation, we can find the final velocity of the plane.
To find the final velocity of the plane, we can use the principle of conservation of linear momentum. According to this principle, the total linear momentum before the missile was fired is equal to the total linear momentum after the missile was fired.
The linear momentum, p, is defined as the product of mass and velocity. So, we can calculate the initial momentum of the system, before the missile was fired, as:
Initial momentum = (mass of plane * velocity of plane) + (mass of missile * velocity of missile relative to plane)
Let's substitute the given values into the equation:
Initial momentum = (5.50 x 10^3 kg * 3.05 x 10^2 m/s) + (1.20 x 10^2 kg * 1.21 x 10^3 m/s)
Next, let's calculate the final momentum, which will be the momentum of the system after the missile was fired. The final momentum can be calculated as:
Final momentum = (mass of plane * final velocity of plane)
We can now set the initial momentum equal to the final momentum and solve for the final velocity of the plane.
(5.50 x 10^3 kg * 3.05 x 10^2 m/s) + (1.20 x 10^2 kg * 1.21 x 10^3 m/s) = (mass of plane * final velocity of plane)
Now, let's rearrange the equation and solve for the final velocity of the plane:
final velocity of plane = [(5.50 x 10^3 kg * 3.05 x 10^2 m/s) + (1.20 x 10^2 kg * 1.21 x 10^3 m/s)] / (mass of plane)
Substituting the given values, we can now calculate the final velocity of the plane.