A fire helicopter carries a 664 kg bucket of water at the end of a 24.9 m long cable. Flying back from a fire at a constant speed of 44.2 m/s, the cable makes an angle of 41.6o with respect to the vertical. Determine the force of air resistance on the bucket.

To determine the force of air resistance on the bucket, we can use Newton's second law of motion, which states that the net force acting on an object is equal to the product of its mass and acceleration:

Fnet = ma

In this case, the net force is equal to the sum of the force of gravity and the force of air resistance:

Fnet = Fgravity + Fair

We can find the force of gravity using the formula:

Fgravity = mg

where m is the mass of the bucket (664 kg) and g is the acceleration due to gravity (approximately 9.8 m/s^2).

Fgravity = (664 kg)(9.8 m/s^2) = 6507.2 N

Now, let's calculate the acceleration of the bucket. We can find the horizontal component of the tension in the cable by calculating the projection of the tension on the horizontal axis:

Thorizontal = Tension * cos(angle)

where Tension is the tension in the cable.

Tension = Tensionvertical / sin(angle)

Now, the vertical component of the tension can be found by:

Tensionvertical = Fgravity + Fair

Therefore,

Tension = (Fgravity + Fair) / sin(angle)

Thorizontal = [(Fgravity + Fair) / sin(angle)] * cos(angle)

Since the helicopter is flying at a constant speed, the acceleration is zero. Therefore:

Fnet = Fgravity + Fair = 0

Substituting the calculated values:

0 = 6507.2 N + Fair

Solving for Fair:

Fair = -6507.2 N

Therefore, the force of air resistance on the bucket is approximately -6507.2 N. The negative sign indicates that the force is in the opposite direction to the force of gravity.