A helicopter rescues a trapped person of mass m = 67.0 kg
from a flooded river by lifting the person vertically upward using a winch and rope. The person is pulled 13.0 m into the helicopter with a constant force that is 11% greater than the person's weight.
(a) Find the work done by each of the forces acting on the person. (Enter your answers to the nearest whole number. Be sure to use 9.81 m/s2 for g.)
Wg =J
WT = J
(b) Assuming the survivor starts from rest, determine his speed upon reaching the helicopter.
m/s
assuming no friction,
work=force*distance=1.13*mg*13
b. if the excess work went into KEnergy, then
kinetic energy=.13*mg*13.0
then KE= 1/2 m v^2 and you can solve for v.
To find the work done by each of the forces acting on the person, we need to calculate the gravitational work (Wg) and the work done by the tension force (WT).
(a) Gravitational work:
The gravitational work done on an object is given by the formula:
Wg = mgh,
where m is the mass of the object, g is the acceleration due to gravity, and h is the height.
In this case, the person is lifted vertically upwards by the helicopter, so the work done against gravity is:
Wg = mg(13.0 m)
→ Wg = (67.0 kg)(9.81 m/s^2)(13.0 m)
→ Wg ≈ 8658 J
The work done by gravity is approximately 8658 Joules.
Tension force work:
The tension force in the rope is 11% greater than the person's weight. We can calculate the tension force (T) as:
T = (1 + 0.11)mg
→ T = 1.11mg
The work done by the tension force is calculated using the formula:
WT = Td,
where T is the tension force and d is the distance over which the force is applied. In this case, the distance is 13.0 m.
Substituting the values:
WT = (1.11mg)(13.0 m)
→ WT ≈ (1.11)(67.0 kg)(9.81 m/s^2)(13.0 m)
→ WT ≈ 9473 J
The work done by the tension force is approximately 9473 Joules.
(b) To determine the survivor's speed upon reaching the helicopter, we can use the work-energy principle, which states that the work done on an object is equal to the change in its kinetic energy.
The total work done on the person is equal to the sum of the work done by gravity and the work done by the tension force:
Total work = Wg + WT
Now, using the work-energy principle, we can equate the work done on the person to the change in kinetic energy:
Total work = ΔKE
In this case, the person starts from rest, so their initial kinetic energy (KEi) is zero. The final kinetic energy (KEf) is given by 0.5mv^2, where v is the final speed.
ΔKE = KEf - KEi
ΔKE = 0.5mv^2 - 0
Equate this to the total work:
Total work = ΔKE
Wg + WT = 0.5mv^2
Substituting the values of Wg and WT:
8658 J + 9473 J = 0.5(67.0 kg)v^2
Simplifying the equation:
18131 J = 33.5v^2
Solving for v:
v^2 = 18131 J / 33.5 kg
v ≈ √(539.88) m/s
v ≈ 23.23 m/s
Therefore, the survivor's speed upon reaching the helicopter is approximately 23.23 m/s.