Answer:
785 N
Step-by-step explanation:
To solve this problem, we can use the principle of conservation of mechanical energy, which states that the total mechanical energy of a system is constant if there are no external forces doing work on the system. In this case, the only force acting on the cyclist is the normal force, which is perpendicular to the surface of the hill and does not do any work. Therefore, the total mechanical energy of the system (cyclist + bicycle) is constant.
The initial total mechanical energy of the system is:
Ei = mgh + (1/2)mv^2
where m is the total mass of the system, g is the acceleration due to gravity, h is the height of the hill, and v is the initial speed of the cyclist.
In this case, the hill has a slope of y = 0.2e^x, so we can use calculus to find the height of the hill:
h = ∫y dx from x = 0 to x = 5
h = ∫(0.2e^x) dx from x = 0 to x = 5
h = 0.2(e^5 - 1)
h ≈ 35.3 m
Substituting the given values, we get:
Ei = (80 kg)(9.81 m/s^2)(35.3 m) + (1/2)(80 kg)(15 m/s)^2
Ei ≈ 30,736 J
The final total mechanical energy of the system is:
Ef = mgh + (1/2)mv^2
where v is the final speed of the cyclist.
Since the cyclist is coasting freely, there is no external force doing work on the system, so the total mechanical energy is conserved:
Ei = Ef
Solving for v, we get:
v = sqrt((2/m)(Ei - mgh))
v = sqrt((2/80 kg)(30,736 J - (80 kg)(9.81 m/s^2)(35.3 m)))
v ≈ 28.5 m/s
The rate of increase in speed is:
v' = (v - 15 m/s)/t
where t is the time interval shown in the problem (not given).
We cannot solve for v' without knowing the value of t.
To find the normal force N, we can use Newton's second law, which states that the net force on an object is equal to its mass times its acceleration:
N - mg = 0
where g is the acceleration due to gravity.
Solving for N, we get:
N = mg
N = (80 kg)(9.81 m/s^2)
N ≈ 785 N
Therefore, the resultant normal force on the bicycle is approximately 785 N.