On a frozen pond, a 13.9 kg sled is given a kick that imparts to it an initial speed of v0 = 1.81 m/s. The coefficient of kinetic friction between sled and ice is μk = 0.122. Use the work kinetic energy theorem to find the distance the sled moves before coming to rest.
call mass m and not 13.9 to save work, it cancels
(1/2) m v^2 = retarding force * d
(1/2)m v^2 = .122 m g d
(1/2)(1.81)^2 = .122(9.81) d
solve for d
Well, this is quite a slippery situation! To find the distance the sled moves before coming to rest, we can use the work kinetic energy theorem.
Now, the work done by friction is equal to the change in kinetic energy. Since the sled comes to rest, the final kinetic energy is zero. Therefore, the work done by friction is equal to the initial kinetic energy.
The initial kinetic energy can be found using the formula KE = (1/2)mv^2, where m is the mass of the sled (13.9 kg) and v is the initial speed (1.81 m/s). Plugging in the values, we have KE = (1/2)(13.9 kg)(1.81 m/s)^2.
Now, the work done by friction can be expressed as W = Fd, where F is the force of friction and d is the distance traveled by the sled before stopping.
The force of friction can be calculated using the equation F = μkN, where μk is the coefficient of kinetic friction and N is the normal force. On a horizontal surface, the normal force is equal to the weight of the sled, which is given by mg, where g is the acceleration due to gravity (approximately 9.8 m/s^2).
Substituting these values, we have F = μkmg. Now, we can substitute this equation into the work done by friction equation, giving W = (μkmg)d.
Since the work done by friction is equal to the initial kinetic energy, we can equate these two equations: (μkmg)d = (1/2)(13.9 kg)(1.81 m/s)^2.
Now, we can solve for d by rearranging the equation: d = [(1/2)(13.9 kg)(1.81 m/s)^2] / (μkmg).
Now, let's plug in the values and calculate the result. But hey, be careful not to slip while doing the math!
To find the distance the sled moves before coming to rest, we can use the work-energy theorem. The work-energy theorem states that the work done on an object equals the change in its kinetic energy.
The work done on the sled is equal to the force applied to it multiplied by the distance it moves. The force of kinetic friction can be calculated as:
frictional force = coefficient of kinetic friction * normal force
The normal force is equal to the weight of the sled, which can be calculated as:
normal force = mass * acceleration due to gravity
The work done by friction is equal to the force of friction multiplied by the distance traveled by the sled before it comes to rest.
At rest, the final velocity of the sled is zero. Therefore, the initial kinetic energy of the sled is equal to the work done by friction.
Using these equations, we can find the distance the sled moves before coming to rest.
Step 1: Calculate the normal force:
mass = 13.9 kg
acceleration due to gravity = 9.8 m/s^2
normal force = mass * acceleration due to gravity
normal force = 13.9 kg * 9.8 m/s^2
normal force = 136.22 N
Step 2: Calculate the force of friction:
coefficient of kinetic friction (μk) = 0.122
frictional force = coefficient of kinetic friction * normal force
frictional force = 0.122 * 136.22 N
frictional force = 16.63 N
Step 3: Calculate the work done by friction:
work done by friction = frictional force * distance
Since the sled comes to rest, the work done by friction will be equal to the initial kinetic energy of the sled.
Step 4: Calculate the initial kinetic energy of the sled:
initial speed (v0) = 1.81 m/s
mass = 13.9 kg
initial kinetic energy = (1/2) * mass * (initial speed)^2
initial kinetic energy = (1/2) * 13.9 kg * (1.81 m/s)^2
initial kinetic energy = 21.05 J
Step 5: Calculate the distance the sled moves before coming to rest:
work done by friction = initial kinetic energy
frictional force * distance = initial kinetic energy
distance = initial kinetic energy / frictional force
distance = 21.05 J / 16.63 N
distance = 1.266 m
Therefore, the sled moves approximately 1.266 meters before coming to rest on the frozen pond.
To find the distance the sled moves before coming to rest, we can use the work kinetic energy theorem. According to this theorem, the work done on an object is equal to the change in its kinetic energy.
The work done on the sled can be calculated using the equation:
Work = Force * Distance
The force acting on the sled is the force of kinetic friction, which can be calculated using the equation:
Force of Friction = μk * Normal Force
The normal force is the force exerted on the sled perpendicular to the surface of the ice. Since the sled is on a horizontal surface, the normal force is equal to the weight of the sled, which can be calculated as:
Normal Force = Mass * Gravity
The acceleration due to gravity, denoted by "g", is approximately 9.8 m/s².
Once we have calculated the force of friction, we can find the work done by multiplying it by the distance the sled moves before coming to rest.
Now, let's calculate the distance the sled moves before coming to rest:
1. Calculate the normal force:
Normal Force = Mass * Gravity
Normal Force = 13.9 kg * 9.8 m/s²
2. Calculate the force of friction:
Force of Friction = μk * Normal Force
Force of Friction = 0.122 * Normal Force
3. Calculate the work done on the sled:
Work = Force of Friction * Distance
According to the work kinetic energy theorem, since the sled starts with an initial speed and comes to rest, the work done on the sled is equal to its initial kinetic energy.
4. Calculate the initial kinetic energy:
Initial Kinetic Energy = (1/2) * Mass * (Initial Velocity)²
Finally, we can use the calculated initial kinetic energy to find the distance the sled moves before coming to rest using the equation:
Distance = Work / Force of Friction
Plug in the values from the problem description into these equations and solve for distance.
initial KE=workdoneonfriction+finalKE
1/2 m vi^2=mu*mg*distance+0
distance=1/2 vi^2/g