A small girl and boy take a sled up to the top of a snow covered slope which is inclined 30 degrees above the horizontal. The girl, boy and the sled have a mass of 160 kg. The coefficient of static friction is 0.28 and the coefficient of kinetic friction is 0.14 between the sled and the snow. They hop on, push off and slide down the 30 meter long incline.
a. How fast are they going at the bottom?
b. How long does it take them to go the 30 meters
To solve this problem, we can use the principles of Newton's laws of motion and the concepts of static and kinetic friction. Here's how we can approach each part of the problem:
a. To find the speed of the sled at the bottom of the slope, we'll need to analyze the forces acting on the sled. At the top of the slope, the only horizontal force acting on the sled is the static friction force, which acts to prevent the sled from sliding down. The maximum static friction force can be calculated using the equation:
Static friction force = coefficient of static friction * normal force
The normal force is the component of the weight acting perpendicular to the slope and can be calculated using:
Normal force = mass * acceleration due to gravity * cos(angle of inclination)
Once we have calculated the static friction force, we can use it to find the acceleration of the sled using Newton's second law:
Sum of forces = mass * acceleration
In this case, the sum of forces is equal to the static friction force. Rearranging the equation, we can solve for acceleration:
acceleration = static friction force / mass
Finally, we can use the equations of motion to find the final velocity at the bottom of the slope. Since the sled starts from rest, the equation relating distance, velocity, acceleration, and time is:
distance = (initial velocity * time) + (0.5 * acceleration * time^2)
Since the sled starts from rest, the initial velocity is zero. Rearranging the equation and solving for velocity (v), we get:
v = sqrt(2 * acceleration * distance)
b. To find the time it takes for them to go the 30 meters, we can use the equation of motion mentioned above:
distance = (initial velocity * time) + (0.5 * acceleration * time^2)
Since the final velocity is unknown, we can solve for time using the quadratic equation:
(time^2 * acceleration / 2) + (initial velocity * time) - distance = 0
Using the quadratic formula, we can solve for time.
Following these steps will allow us to find the answers to both parts of the problem.