A 56.9 kg sledder descends an 8.21-meter high hill, encountering a friction force of 11.7 N, Fill in the blanks and determine the speed of the sledder after traveling the 31.7 meters to the bottom of the hill.
Top of the hill: KE=0J PE=_ W=_
Work=_
PE=0J KE=_ At the bottom of the hill
height=8.21M
At the top of hill:
PE = M*g*h = 56.9 * 9.8 * 8.21 = 4578 J.
At bottom of hill:
KE = PE - Fk*d.
0.5M*V^2 = 4578 - 11.7*31.7
0.5*56.9*V^2 = 4207,
V^2 = 148, V = 12.2 m/s.
Well, isn't that a slippery situation! Let's calculate the energy changes as our sledder races down the hill.
At the top of the hill, the kinetic energy (KE) is 0J because they haven't started moving yet, and the potential energy (PE) is mgh, which means PE = 56.9kg * 9.8m/s^2 * 8.21m, which equals approximately 4484.3894 Joules (let's round it to 4484J). Since there is friction, the work done (W) is equal to the friction force multiplied by the distance traveled, which is 11.7N * 31.7m, yielding about 370.89 Joules (roughly 371J).
Once our brave sledder reaches the bottom of the hill, the potential energy is now 0J because they're at ground level, and the kinetic energy can be calculated using the law of conservation of energy: KE = PE + W. So KE = 4484J + 371J, resulting in approximately 4855 Joules. Now we can determine the speed using the formula KE = 1/2 mv^2. Plugging in the values, we get 4855J = (1/2) * 56.9kg * v^2. Solving for v, we find v ≈ 10.02 m/s (rounding to two decimal places).
So, after sledding down the hill, our sledder ends up with a speed of approximately 10.02 m/s. Watch out for snowballs!
To determine the speed of the sledder at the bottom of the hill, we can use the conservation of mechanical energy.
The initial mechanical energy at the top of the hill consists of both kinetic energy (KE) and potential energy (PE).
Given:
Mass of the sledder (m) = 56.9 kg
Height of the hill (h) = 8.21 m
Friction force (F) = 11.7 N
The initial potential energy (PE) at the top of the hill can be calculated using the formula:
PE = m * g * h
where g is the acceleration due to gravity (approximately 9.8 m/s²).
PE = 56.9 kg * 9.8 m/s² * 8.21 m
PE ≈ 4371.2242 J
At the top of the hill, the kinetic energy is 0 J since the sledder is starting from rest (KE = 0 J).
Now, let's calculate the work done by friction:
Work = F * d
where d is the distance traveled (31.7 m).
Work = 11.7 N * 31.7 m
Work ≈ 370.89 J
As per the conservation of mechanical energy, the work done by friction is equal to the change in mechanical energy:
Work = KE_final - KE_initial
Since the kinetic energy (KE) was 0 J at the top of the hill, we can rearrange the equation:
KE_final = KE_initial + Work
KE_final = 0 J + 370.89 J
KE_final ≈ 370.89 J
At the bottom of the hill, the potential energy (PE) is 0 J since the sledder has descended to the lowest point and all the potential energy has converted to kinetic energy.
Now we can calculate the final kinetic energy:
KE_final = PE
370.89 J = m * g * h
m * g * h = 370.89 J
m * 9.8 m/s² * 8.21 m = 370.89 J
m ≈ 370.89 J / (9.8 m/s² * 8.21 m)
m ≈ 4.5 kg
So, the speed of the sledder at the bottom of the hill can be calculated using the formula:
KE = (1/2) * m * v^2
where KE is the final kinetic energy and m is the mass of the sledder.
370.89 J = (1/2) * 4.5 kg * v^2
v^2 ≈ (2 * 370.89 J) / (4.5 kg)
v^2 ≈ 164.6422 m²/s²
Taking the square root of both sides to find the speed (v):
v ≈ √(164.6422 m²/s²)
v ≈ 12.84 m/s
Therefore, the speed of the sledder after traveling the 31.7 meters to the bottom of the hill is approximately 12.84 m/s.
To solve this problem, we need to use the principle of conservation of mechanical energy. The total mechanical energy of the system remains constant when only non-conservative forces (such as friction) are present. In this case, the initial potential energy (PE) at the top of the hill will be converted into kinetic energy (KE) at the bottom of the hill, assuming no energy losses.
Let's fill in the blanks step by step:
Top of the hill:
- KE = 0 J (initially at rest)
- PE = ?
- W = ?
At the top of the hill, all the energy is in the form of potential energy. So, the potential energy at the top of the hill is calculated using the equation:
PE = m * g * h
where:
m = mass of the sledder = 56.9 kg
g = acceleration due to gravity = 9.8 m/s^2
h = height of the hill = 8.21 m
Substituting the values, we get:
PE = 56.9 kg * 9.8 m/s^2 * 8.21 m
Now, calculate the potential energy.
PE = 4555.2382 J (rounded to four decimal places)
Next, let's calculate the work done by the friction force. The work done by a force is given by the equation:
Work = force * distance
In this case, the friction force is given as 11.7 N, and the sledder travels a distance of 31.7 meters. So, the work done by the friction force is:
Work = 11.7 N * 31.7 m
Now, calculate the work done:
Work = 370.89 J (rounded to two decimal places)
Now, let's move to the bottom of the hill and calculate the kinetic energy (KE) and speed of the sledder.
At the bottom of the hill:
- PE = 0 J (all converted into KE)
- KE = ?
- height = 8.21 m (as given)
Since the potential energy (PE) is converted into kinetic energy (KE), we can use the equation:
PE + KE = Constant
Since PE = 0 J, we can simplify it to:
KE = Constant
So, the kinetic energy at the bottom of the hill is the same as the potential energy at the top of the hill.
KE = 4555.2382 J (rounded to four decimal places)
Finally, we can calculate the speed (v) of the sledder using the formula:
KE = (1/2) * m * v^2
Rearrange the equation to solve for v:
v = sqrt(2 * KE / m)
Substituting the values, we get:
v = sqrt(2 * 4555.2382 J / 56.9 kg)
Now, calculate the speed:
v ≈ 12.648 m/s (rounded to three decimal places)
Therefore, the speed of the sledder after traveling 31.7 meters to the bottom of the hill is approximately 12.648 m/s.