A girl starting from rest at the top of a 40 m jump, inclined at 30º above the horizontal, has 0.15 coefficient of friction. The jump ends with a frictionless, horizontal part. How far does she land away from the jump, which is 50 m above the horizontal land below?
To solve this problem, we can break it down into two parts: the jump and the landing.
First, let's calculate the velocity of the girl when she reaches the end of the 40 m jump. We can use the conservation of energy principle.
The initial potential energy at the top of the jump is given by:
PE_initial = m * g * h
where m is the mass of the girl, g is the acceleration due to gravity (approximately 9.8 m/s^2), and h is the height of the jump (40 m).
Since the girl starts from rest, her initial kinetic energy is zero:
KE_initial = 0
At the end of the jump, the potential energy is zero because the girl is at ground level. Therefore, all the initial potential energy is converted into kinetic energy:
PE_final = 0
KE_final = (1/2) * m * v^2
where v is the velocity of the girl at the end of the jump.
Using the principle of conservation of energy, we have:
PE_initial + KE_initial = PE_final + KE_final
m * g * h + 0 = 0 + (1/2) * m * v^2
Simplifying the equation:
m * g * h = (1/2) * m * v^2
g * h = (1/2) * v^2
Solving for v:
v^2 = 2 * g * h
v = sqrt(2 * g * h)
Now, let's calculate the velocity:
v = sqrt(2 * 9.8 * 40)
v = 28 m/s (rounded to two decimal places)
Next, we can calculate the distance the girl lands away from the jump. Since there is no friction during the landing, we can use horizontal motion equations.
First, let's calculate the time taken for the girl to fall from the jump height, which is 50 m above the horizontal land below. We can use the equation of motion:
h = (1/2) * g * t^2
50 = (1/2) * 9.8 * t^2
t^2 = 50 / (1/2 * 9.8)
t^2 = 50 / 4.9
t^2 = 10.2
t ≈ 3.20 seconds (rounded to two decimal places)
Now, let's calculate the horizontal distance traveled during this time:
d = v * t
d = 28 * 3.20
d ≈ 89.6 meters (rounded to one decimal place)
Therefore, the girl lands approximately 89.6 meters away from the jump.