On a frozen pond, a person kicks a 10.0kg sled, giving it an initial speed of 2.2m/s. How far does the sled move if the coefficient of kinetic friction between the sled and the ice is 0.10?
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To determine how far the sled moves, we can start by calculating the force of kinetic friction acting on the sled.
The equation for the force of kinetic friction is given by:
F_k = μ_k * N
Where:
F_k is the force of kinetic friction
μ_k is the coefficient of kinetic friction
N is the normal force
In this case, the normal force is equal to the weight of the sled, which can be calculated using the equation:
N = m * g
Where:
m is the mass of the sled
g is the acceleration due to gravity (approximately 9.8 m/s^2)
Given that the mass of the sled (m) is 10.0 kg, the acceleration due to gravity (g) is 9.8 m/s^2, and the coefficient of kinetic friction (μ_k) is 0.10, we can now calculate the force of kinetic friction:
F_k = 0.10 * (10.0 kg * 9.8 m/s^2)
F_k = 0.10 * 98 N
F_k = 9.8 N
Next, we can use the work-energy principle to determine the distance the sled moves. The work done on an object is equal to the change in its kinetic energy. The work done against friction can be calculated using the equation:
W = F * d
Where:
W is the work done against friction
F is the force of kinetic friction
d is the distance traveled by the sled
The work done against friction is equal to the change in kinetic energy:
W = ΔKE
The initial kinetic energy of the sled, K1, can be calculated using the equation:
K1 = 1/2 * m * v1^2
Where:
K1 is the initial kinetic energy
m is the mass of the sled
v1 is the initial velocity of the sled
Given that the mass of the sled (m) is 10.0 kg and the initial velocity of the sled (v1) is 2.2 m/s, we can calculate the initial kinetic energy:
K1 = 1/2 * 10.0 kg * (2.2 m/s)^2
K1 = 1/2 * 10.0 kg * 4.84 m^2/s^2
K1 = 24.2 J
The final kinetic energy of the sled, K2, is equal to zero because it eventually comes to a stop:
K2 = 0 J
Since the work done against friction is equal to the change in kinetic energy, we have:
W = ΔKE
W = K2 - K1
W = 0 J - 24.2 J
W = -24.2 J
Since the work done against friction is equal to the force of kinetic friction multiplied by the distance traveled, we can rearrange the equation to solve for the distance traveled:
W = F * d
-24.2 J = 9.8 N * d
Solving for d:
d = -24.2 J / 9.8 N
d ≈ -2.47 m
The negative sign indicates that the sled moves in the opposite direction of the initial kick. Therefore, the sled moves approximately 2.47 meters in the opposite direction.