A 55 kg skier is at the top of a slope. The vertical distance between start and finish is 10m. (a) What is her Potential Energy? (b) How fast is she moving at the bottom of the hill?
To answer these questions, we need to apply the concepts of potential energy and conservation of energy.
(a) The potential energy (PE) of an object is given by the equation PE = mgh, where m is the mass of the object, g is the acceleration due to gravity (9.8 m/s²), and h is the height or vertical distance.
Plugging in the values into the formula:
m = 55 kg
g = 9.8 m/s²
h = 10 m
PE = (55 kg) * (9.8 m/s²) * (10 m)
PE = 5390 Joules
So, the potential energy of the skier is 5390 Joules.
(b) To find the speed of the skier at the bottom of the hill, we can use the principle of conservation of energy. Assuming there is no energy lost to friction or air resistance, the initial potential energy (PE) is converted entirely into kinetic energy (KE) at the bottom.
The equation for kinetic energy is KE = 0.5 * m * v², where m is the mass of the object and v is its velocity.
Since the potential energy is converted into kinetic energy, we can equate them:
PE = KE
mgh = 0.5mv²
Simplifying the equation:
mgh = 0.5mv²
gh = 0.5v²
v² = 2gh
Plugging in the values:
m = 55 kg
g = 9.8 m/s²
h = 10 m
v² = 2 * (9.8 m/s²) * (10 m)
v² = 196 m²/s²
v ≈ 14 m/s (taking the square root)
Therefore, the skier would be moving at approximately 14 m/s at the bottom of the hill.
To find the potential energy of the skier at the top of the slope, we can use the formula:
Potential Energy = mass * gravity * height
where
mass = 55 kg (given)
gravity = 9.8 m/s² (acceleration due to gravity)
height = 10 m (given)
(a) Calculating the potential energy using the formula:
Potential Energy = 55 kg * 9.8 m/s² * 10 m
= 5390 J (Joules)
Therefore, the potential energy of the skier at the top of the slope is 5390 Joules.
To find the speed of the skier at the bottom of the hill, we can use the principle of conservation of energy, where the potential energy is converted to kinetic energy.
The total mechanical energy of the skier is given by the sum of potential energy and kinetic energy:
Total Mechanical Energy = Potential Energy + Kinetic Energy
At the top of the hill, all the energy is potential energy, and at the bottom, all the energy is kinetic energy. Therefore:
Total Mechanical Energy at the top = Potential Energy
Total Mechanical Energy at the bottom = Kinetic Energy
The formula for kinetic energy is:
Kinetic Energy = 0.5 * mass * velocity²
where
mass = 55 kg (given)
velocity = unknown, to be determined
(b) Since all the potential energy at the top is converted into kinetic energy at the bottom, we can equate the two:
Potential Energy = Kinetic Energy
Potential Energy at the top = Kinetic Energy at the bottom
Using the formula for kinetic energy:
Potential Energy at the top = 0.5 * mass * velocity²
Substituting the given values:
5390 J = 0.5 * 55 kg * velocity²
Simplifying:
5390 J = 27.5 kg * velocity²
Dividing both sides by 27.5 kg:
velocity² = 5390 J / 27.5 kg
velocity² ≈ 196 m²/s²
Taking the square root of both sides:
velocity ≈ √(196 m²/s²)
velocity ≈ 14 m/s
Therefore, the skier is moving at a speed of approximately 14 m/s at the bottom of the hill.
(a) P.E. = m g h
(b) neglecting friction (slippery snow)
... the P.E. at the top becomes K.E. at the bottom
... m g h = 1/2 m v^2
... v = √(2 g h)