A girl on a sled coasts down a hill. Her speed is 6.0 m/s when she reaches level ground at the bottom. The coefficient of kinetic friction between the sled's runners and the hard, icy snow is 0.053, and the girl and sled together weigh 674 N. How far does the sled travel on the level ground before coming to rest?
To find the distance the sled travels on the level ground before coming to rest, we can use the equation for the force of kinetic friction.
The force of kinetic friction can be calculated using the formula:
frictional force = coefficient of kinetic friction * normal force
The normal force can be determined by multiplying the weight of the sled and the girl by the acceleration due to gravity (9.8 m/s^2).
normal force = weight * gravity
Now, we need to find the initial kinetic energy of the sled.
The formula for kinetic energy is:
kinetic energy = 0.5 * mass * velocity^2
By rearranging this formula, we can find the mass:
mass = kinetic energy / (0.5 * velocity^2)
Substituting the values into the equation:
mass = (0.5 * (674 N) * (6.0 m/s)^2) / (0.5 * (9.8 m/s^2))
= (0.5 * 674 * 36) / 4.9
= 3097.8 kg (approximately)
Now, we can determine the frictional force:
frictional force = (coefficient of kinetic friction * normal force)
= (0.053 * (674 N * 9.8 m/s^2))
= 345.434 N (approximately)
Since the sled comes to rest, the force of kinetic friction is equal to the force pushing the sled forward.
Now, we can use the equation for force:
force = mass * acceleration
The acceleration in this case is equal to (the force of kinetic friction) / mass:
acceleration = frictional force / mass
= 345.434 N / 3097.8 kg
= 0.1115 m/s^2
Since the sled comes to rest, its final velocity is 0 m/s. We can use the equation to calculate the distance:
velocity^2 = initial velocity^2 + 2 * acceleration * distance
Rearranging this formula to solve for distance:
distance = (velocity^2 - initial velocity^2) / (2 * acceleration)
= (0 - (6.0 m/s)^2) / (2 * (-0.1115 m/s^2))
= 16.2 m (approximately)
Therefore, the sled travels approximately 16.2 meters on the level ground before coming to rest.