A child and sled with a combined mass of 47.8 kg slide down a frictionless hill that is 7.18 m high. If the sled starts from rest, what is its speed at the bottom of the hill?
V^2 = Vo^2 + 2g*h.
V^2 = 0 + 19.6*7.18 = 140.738
V = 11.86 m/s.
To find the speed of the sled at the bottom of the hill, we can use the principle of conservation of energy. The potential energy at the top of the hill is transformed into kinetic energy at the bottom of the hill.
The potential energy (PE) at the top of the hill can be calculated using the formula:
PE = m * g * h
where m is the mass, g is the acceleration due to gravity (around 9.8 m/s^2), and h is the height of the hill.
PE = 47.8 kg * 9.8 m/s^2 * 7.18 m
PE = 3304.8848 J
This potential energy is converted into kinetic energy (KE) at the bottom of the hill.
KE = 1/2 * m * v^2
where m is the mass and v is the velocity.
Let's calculate the velocity:
KE = 3304.8848 J
1/2 * 47.8 kg * v^2 = 3304.8848 J
Multiplying both sides of the equation by 2:
47.8 kg * v^2 = 6609.7696 J
Dividing both sides of the equation by 47.8 kg:
v^2 = 138.2597 m^2/s^2
To solve for v, we take the square root of both sides:
v = sqrt(138.2597 m^2/s^2)
v ≈ 11.75 m/s
Therefore, the sled's speed at the bottom of the hill is approximately 11.75 m/s.
To find the speed of the sled at the bottom of the hill, we can use the principle of conservation of energy.
The potential energy (PE) at the top of the hill is converted into kinetic energy (KE) at the bottom, as there is no friction involved. The equation for potential energy is given by:
PE = mgh
Where:
m = mass of the system (child + sled) = 47.8 kg
g = acceleration due to gravity = 9.8 m/s^2
h = height of the hill = 7.18 m
The equation for kinetic energy is given by:
KE = 0.5mv^2
Where:
v = velocity, or speed, at the bottom of the hill (what we want to find)
Using the conservation of energy principle, we can equate the potential energy at the top of the hill to the kinetic energy at the bottom:
PE = KE
mgh = 0.5mv^2
Canceling out the mass, we get:
gh = 0.5v^2
Now we can plug in the values and solve for v:
(9.8 m/s^2)(7.18 m) = 0.5v^2
v^2 = (2 * 9.8 m/s^2 * 7.18 m)
v^2 = 138.4884
v = √(138.4884)
v ≈ 11.77 m/s
Therefore, the speed of the sled at the bottom of the hill is approximately 11.77 m/s.