Some children go tobogganing on an icy hill. The start from the rest at the top of the hill. The toboggan and children have a combined mass of 94 kg. The height of the first hill a is 12.0m the height of b is 0, and the height of c is 3.0m. If friction is small enough to be ignored , apply the law of conservation of energy to determine
A) the total mechanical energy of the toboggan at a relative to b
B) the sped of the toboggan at b
C) the speed of the toboggan at c
E = (1/2) m v^2 + m g h = constant forever in this problem
mass does not matter here, cancels
E = m g (12) forever
m g (12) = (1/2) m v^2 + m g(0) at b
12 * 9.8 = (1/2) v^2 solve for v at b
m g (12) = (1/2) m v^2 + m g (3)
9 * 9.8 = (1/2) m v^2 solve for v at c
To answer these questions, we will apply the law of conservation of energy, which states that the total mechanical energy of a system remains constant in the absence of external forces. In this case, we can use the following equation to calculate the mechanical energy:
Total Mechanical Energy = Potential Energy + Kinetic Energy
First, let's calculate the mechanical energy at point a:
A) The total mechanical energy of the toboggan at point a (relative to point b) can be calculated as follows:
Mechanical Energy at a = Potential Energy at a + Kinetic Energy at a
The potential energy at a can be calculated using the formula: Potential Energy = mass * acceleration due to gravity * height
Potential Energy at a = (mass of toboggan and children) * (acceleration due to gravity) * (height of a)
Potential Energy at a = 94 kg * 9.8 m/s^2 * 12.0 m
Next, let's calculate the kinetic energy at point a.
Since the children and the toboggan are initially at rest, the kinetic energy at a is zero.
Hence, the total mechanical energy at point a (relative to point b) is equal to the potential energy at a:
Total Mechanical Energy at a = Potential Energy at a
Now, let's move on to point b:
B) We are tasked to find the speed of the toboggan at point b.
At point b, all the potential energy at a is converted into kinetic energy since the height is zero.
Using the conservation of energy principle, we can equate the potential energy at a to the kinetic energy at b:
Potential Energy at a (height) = Kinetic Energy at b
Now, substitute the potential energy at a (calculated in part A) with the kinetic energy at b and solve for the speed.
Potential Energy at a = Kinetic Energy at b
(94 kg * 9.8 m/s^2 * 12.0 m) = (0.5 * mass of toboggan * velocity at b^2)
Solve the equation to find the velocity at b.
C) Now, we need to find the speed of the toboggan at point c.
At point c, both the potential energy and kinetic energy are at their minimum since the height is 3.0m from the ground.
From the conservation of energy principle, we know that:
Potential Energy at c + Kinetic Energy at c = Mechanical Energy at a
Since the potential energy at c is zero (the height is 3.0m), we can rearrange the equation as follows:
Kinetic Energy at c = Mechanical Energy at a
Substitute the mechanical energy at a (calculated in part A) with the kinetic energy at c and solve for the speed.
Kinetic Energy at c = Mechanical Energy at a
Finally, solve the equation to find the speed at point c.
By following these steps, we can find the answers to parts A, B, and C using the law of conservation of energy.