(a) Well, the arresting cables do quite a bit of work, but they are not exactly getting a promotion to "Cable of the Year." To find the work done, we can use the equation W = Fd, where W is the work, F is the force, and d is the distance.
The distance here is 84 m, but we need to convert the force to Newtons. Since 1 kilonewton (kN) = 1000 N, we find that the force is 220,000 N. Now, we can plug in the values into our equation:
W = (220,000 N) * (84 m)
W = 18,480,000 Nm
W = 18,480,000 Joules
So, the arresting cables do approximately 18,480,000 Joules of work on the plane. That's quite a workout!
(b) To find the force exerted on the plane by the cables, we can use the equation F = ma, where F is the force, m is the mass, and a is the acceleration. Since the plane is brought to a stop, its final velocity is 0 m/s.
First, let's convert the mass from tons to kilograms. We know that 1 ton is equal to 1000 kg, so the mass of the plane is 25 * 1000 kg = 25,000 kg.
Next, we can use the equation v^2 = u^2 + 2ad, where v is the final velocity, u is the initial velocity, a is the acceleration, and d is the distance. Rearranging the equation to solve for acceleration, we get a = (v^2 - u^2) / (2d).
Plugging in the values, we have:
a = (0^2 - (67 m/s)^2) / (2 * 84 m)
a = (-4489 m^2/s^2) / (168 m)
a ≈ -26.7 m/s^2 (approximately)
Now, we can find the force:
F = (25,000 kg) * (-26.7 m/s^2)
F ≈ -667,500 N
So, the force exerted on the plane by the cables is approximately -667,500 N. Hmm, negative force? Seems like those cables need a lesson in being more positive!