A child and sled with a combined mass of 50.0 kg slide down a frictionless hill that is 7.34 m high. If the sled starts from rest, what is the mechanical energy of the system, and what is the sled’s speed at the bottom of the hill?
m g h = 50 * 9.81 * 7.34 = 3600 Joules
Ke at bottom = (1/2) m v^2 = 3600 Joules
50 v^2 = 7200
v^2 = 144
v = 12 m/s
Well, that's a slippery situation you've got there! Let me calculate the mechanical energy and speed for you.
The mechanical energy of the system at the top of the hill can be found using the formula:
E = mgh
where E is the mechanical energy, m is the mass (50.0 kg), g is the acceleration due to gravity (9.8 m/s^2), and h is the height of the hill (7.34 m). Plugging in the values:
E = (50.0 kg) × (9.8 m/s^2) × (7.34 m)
E = 359.98 J
So, the mechanical energy of the system at the top of the hill is approximately 359.98 Joules. Now, to calculate the sled's speed at the bottom of the hill, we can use the conservation of mechanical energy:
E = (1/2)mv²
where v is the speed of the sled at the bottom of the hill. Rearranging the equation to solve for v:
v = sqrt((2E) / m)
v = sqrt((2 × 359.98 J) / 50.0 kg)
v = sqrt(14.3996 m²/s²)
v ≈ 3.798 m/s
So, the sled's speed at the bottom of the hill is approximately 3.798 m/s.
Looks like that sled is sliding down the hill faster than a greased-up penguin! Enjoy the ride!
To find the mechanical energy of the system, we need to consider the potential energy and the kinetic energy.
Potential Energy (PE) is given by the formula PE = m * g * h, where m is the mass, g is the acceleration due to gravity (9.8 m/s^2), and h is the height.
In this case, the potential energy at the top of the hill is PE = 50.0 kg * 9.8 m/s^2 * 7.34 m.
PE = 3549.16 Joules
At the bottom of the hill, all the potential energy is converted into kinetic energy.
Kinetic Energy (KE) is given by the formula KE = 0.5 * m * v^2, where m is the mass and v is the velocity.
So, we need to solve the equation PE = KE to find the velocity (v).
3549.16 = 0.5 * 50.0 kg * v^2
Dividing both sides by 0.5 * 50.0 kg, we get:
v^2 = (3549.16 / (0.5 * 50.0 kg))
v^2 = 141.9664 m^2/s^2
Taking the square root of both sides, we get:
v ≈ 11.91 m/s
Therefore, the sled's speed at the bottom of the hill is approximately 11.91 m/s.
To find the mechanical energy of the system, we can use the principle of conservation of energy. The mechanical energy of an object is the sum of its potential energy and kinetic energy.
The potential energy of an object is given by the formula PE = mgh, where m is the mass, g is the acceleration due to gravity, and h is the height.
In this case, the potential energy of the system is equal to the potential energy of the sled and the child. So, the potential energy is PE = (m_sled + m_child) * g * h, where m_sled is the mass of the sled, m_child is the mass of the child, g is the acceleration due to gravity, and h is the height of the hill.
The kinetic energy of an object is given by the formula KE = (1/2) * m * v^2, where m is the mass and v is the velocity.
Since the system starts from rest, the initial kinetic energy is zero.
According to the principle of conservation of energy, the mechanical energy of the system is conserved, so the total mechanical energy at the top of the hill (potential energy) is equal to the total mechanical energy at the bottom of the hill (potential energy + kinetic energy).
So, we can set up the equation:
(PE)_top = (PE)_bottom + (KE)_bottom
Let's substitute the formulas for potential energy and kinetic energy:
(m_sled + m_child) * g * h = (m_sled + m_child) * g * 0 + (1/2) * (m_sled + m_child) * v^2
Now we can solve for the sled's speed at the bottom of the hill (v):
v^2 = 2 * g * h
v = sqrt(2 * g * h)
The value of the acceleration due to gravity (g) can be taken as approximately 9.8 m/s^2.
Now we can substitute the values given in the question:
h = 7.34 m
g = 9.8 m/s^2
v = sqrt(2 * 9.8 m/s^2 * 7.34 m)
After evaluating this expression, we can find the sled's speed at the bottom of the hill.
To find the mechanical energy of the system, we substitute the values of mass and height into the potential energy formula:
PE = (m_sled + m_child) * g * h
After evaluating this expression, we can find the mechanical energy of the system.