An attack helicopter is equipped with a 20-
mm cannon that fires 61.6 g shells in the
forward direction with a muzzle speed of
585 m/s. The fully loaded helicopter has a
mass of 4890 kg. A burst of 208 shells is fired
in a 2.32 s interval.
What is the resulting average force on the
helicopter?
Answer in units of N
By what amount is its forward speed reduced?
Answer in units of m/s
To find the resulting average force on the helicopter, we can use Newton's second law of motion, which states that the force an object experiences is equal to the rate of change of its momentum.
To calculate the resulting average force, we need to first determine the change in momentum of the helicopter caused by the burst of shells.
The momentum of an object is given by the product of its mass and velocity. In this case, the momentum change is due to the firing of the shells, so we need to calculate the momentum change for each shell fired, and then sum them up.
The momentum change for each shell can be found using the formula:
Δp = mΔv
where Δp is the change in momentum, m is the mass of the shell, and Δv is the change in velocity (which is the muzzle speed, since the shell is initially at rest).
In this case, the mass of each shell is given as 61.6 g (or 0.0616 kg), and the muzzle speed is given as 585 m/s. So the change in momentum for each shell is:
Δp = (0.0616 kg)(585 m/s) = 36.036 kg·m/s
Now, since a burst of 208 shells is fired, the total change in momentum is:
Δptotal = (208)(36.036 kg·m/s) = 7480.128 kg·m/s
Since force is the rate of change of momentum, we can find the average force by dividing the total change in momentum by the time interval over which it occurs:
Favg = Δptotal / Δt
In this case, the time interval is given as 2.32 s. So the average force exerted on the helicopter is:
Favg = 7480.128 kg·m/s / 2.32 s = 3224 N
Therefore, the resulting average force on the helicopter is 3224 N.
To determine how much the forward speed of the helicopter is reduced, we can apply the principle of conservation of momentum. In the absence of external forces, the total momentum of a system remains constant.
The initial momentum of the helicopter can be calculated by multiplying its mass and initial velocity:
Pi = mass of the helicopter * initial velocity
Pi = 4890 kg * 0 m/s (since the helicopter is initially at rest)
Pi = 0 kg·m/s
The final momentum of the helicopter can be calculated by multiplying its updated mass (taking into account the shells that are fired) and the resulting speed:
Pf = updated mass of the helicopter * updated velocity
The updated mass of the helicopter is the initial mass plus the mass of the fired shells:
Updated mass = mass of the helicopter + mass of shells fired
Updated mass = 4890 kg + (208 shells * 0.0616 kg/shell)
Updated mass = 4890 kg + 12.84 kg
Updated mass = 4902.84 kg
The final velocity of the helicopter can be determined by dividing the change in momentum (Δptotal) by the updated mass of the helicopter:
Pf = Δptotal / updated mass
Pf = 7480.128 kg·m/s / 4902.84 kg
Pf ≈ 1.526 m/s
Therefore, the forward speed of the helicopter is reduced by approximately 1.526 m/s.