A child and sled with a combined mass of 50.0 kg slide down a frictionless slope. If the sled starts from rest and has a speed of 3.40 m/s at the bottom, what is the height of the hill?

m g h = (1/2) m v^2

as you can see the mass has no effect

h = v^2/(2g)

To find the height of the hill, we can use the law of conservation of energy. The total mechanical energy at the top is equal to the total mechanical energy at the bottom of the hill.

The total mechanical energy at the top consists of the potential energy (PE) and the kinetic energy (KE) due to the initial velocity of the sled:

Initial potential energy (PE1) = mgh, where m is the mass and g is the acceleration due to gravity (approximately 9.8 m/s²).

The total mechanical energy at the bottom consists of the potential energy (PE) and the kinetic energy (KE) due to the final velocity of the sled:

Final kinetic energy (KE2) = (1/2)mv², where v is the final velocity of the sled.

Since the sled starts from rest, the initial kinetic energy (KE1) is zero.

Therefore, according to the law of conservation of energy:

PE1 + KE1 = PE2 + KE2
mgh + 0 = 0 + (1/2)mv²

Since the mass of the sled and child combined is 50.0 kg, and the final velocity v is 3.40 m/s, we can rearrange the equation to solve for h:

50.0 kg * 9.8 m/s² * h = (1/2) * 50.0 kg * (3.40 m/s)²

Simplifying the equation:

490h = (1/2) * 50 * (3.40)²
490h = 340

Now, divide both sides of the equation by 490 to solve for h:

h = 340 / 490
h ≈ 0.694 m

Therefore, the height of the hill is approximately 0.694 meters.

To determine the height of the hill, we can use the principle of conservation of energy. When the sled is at the top of the hill, it has only gravitational potential energy (PE), and when it reaches the bottom, it only has kinetic energy (KE). Since we neglect friction, the total mechanical energy of the system remains constant.

The potential energy (PE) and kinetic energy (KE) can be calculated using the following formulas:

PE = m * g * h
KE = (1/2) * m * v^2

Where:
m is the mass of the sled and child (50.0 kg in this case)
g is the acceleration due to gravity (approximately 9.8 m/s^2)
h is the height of the hill (what we're trying to find)
v is the final velocity at the bottom of the hill (3.40 m/s in this case)

Since the sled starts from rest, the initial velocity (u) is 0 m/s.

Using the principle of conservation of energy, we can equate the potential energy at the top of the hill with the kinetic energy at the bottom of the hill:

PE = KE

m * g * h = (1/2) * m * v^2

Canceling out the mass (m) on both sides of the equation:

g * h = (1/2) * v^2

Rearranging the equation to solve for h:

h = (1/2) * (v^2 / g)

Plugging in the given values:

h = (1/2) * (3.40^2 / 9.8)

Calculating:

h ≈ (1/2) * (11.56 / 9.8)
h ≈ (1/2) * 1.18
h ≈ 0.59 meters

Therefore, the height of the hill is approximately 0.59 meters.

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