A toboggan approaches a snowy hill moving at 15.0 m/s. The coefficients of static and kinetic friction between the snow and the toboggan are 0.460 and 0.350, respectively, and the hill slopes upward at 41.0 above the horizontal. Find the acceleration of the toboggan as it is going up hill, and find the acceleration after it has reached the highest point and is sliding down hill.
To find the acceleration of the toboggan as it is going uphill, we need to consider the forces acting on it.
1. First, let's find the force of gravity pulling the toboggan down the slope. The force of gravity can be calculated using the formula:
Force of gravity = mass of the toboggan × acceleration due to gravity
Given that the uphill slope is 41.0 degrees above the horizontal, the component of gravity acting down the slope can be found using trigonometry:
Force of gravity down the slope = Force of gravity × sin(41.0)
2. Next, let's calculate the force of static friction acting uphill. The force of static friction can be calculated using the formula:
Force of static friction = coefficient of static friction × normal force
The normal force can be found using the component of gravity perpendicular to the slope:
Normal force = Force of gravity × cos(41.0)
Finally, multiply the coefficient of static friction by the normal force to get the force of static friction.
3. Now, let's find the net force acting uphill. The net force is the difference between the force of static friction uphill and the force of gravity down the slope:
Net force uphill = Force of static friction - Force of gravity down the slope
4. Finally, use Newton's second law of motion to find the acceleration uphill:
Net force uphill = mass of the toboggan × acceleration uphill
Rearranging the formula, we get:
Acceleration uphill = Net force uphill / mass of the toboggan
To find the acceleration after the toboggan has reached the highest point and is sliding downhill, we need to consider the forces acting on it.
1. The only force acting on the toboggan going downhill is the force of kinetic friction. The force of kinetic friction can be calculated using the formula:
Force of kinetic friction = coefficient of kinetic friction × normal force
Use the same normal force value as calculated earlier.
2. The net force acting downhill is the force of kinetic friction.
3. Finally, use Newton's second law of motion to find the acceleration downhill:
Net force downhill = mass of the toboggan × acceleration downhill
Rearranging the formula, we get:
Acceleration downhill = Net force downhill / mass of the toboggan
By using the given information and applying the above steps, you can find the acceleration uphill and downhill for the toboggan.