A boy coasts down a hill on a sled, reaching a level surface at the bottom with a speed of 6.2 m/s. If the coefficient of friction between the sled's runners and snow is 0.060 and the boy and sled together weigh 480 N, how far does the sled travel on the level surface before coming to rest?
Oh, it sounds like the boy and sled are going on a chilly adventure! Let's calculate how far they travel before coming to a stop.
To start, we need to find the force of friction acting on the sled. The formula for friction is:
Friction = coefficient of friction * normal force
Given that the coefficient of friction is 0.060 and the weight of the boy and sled is 480 N, we can calculate the frictional force as:
Friction = 0.060 * 480 N
After calculating that, we find that the frictional force is 28.8 N.
Now, to find the distance traveled before stopping, we use the formula:
f = 0.5 * m * v^2
Where f is the force of friction, m is the mass, and v is the velocity.
Since the mass is not given, we can use the weight divided by the acceleration due to gravity (480 N / 9.8 m/s^2) to find the mass. After calculating that, we have:
Mass = 48.98 kg
Now we can rearrange the formula to solve for distance (d):
Distance = (v^2) / (2 * (f / m))
Plugging in the values, we get:
Distance = (6.2 m/s)^2 / (2 * (28.8 N / 48.98 kg))
After evaluating that, we find that the sled will travel approximately 1.33 meters before coming to a stop. That's not bad for a snowy adventure!
To find the distance the sled travels before coming to rest, we need to know the deceleration caused by the friction force. We can find it using the equation:
friction force = coefficient of friction × normal force
Where:
- friction force is the opposing force due to friction
- coefficient of friction is given as 0.060
- normal force is the force exerted perpendicular to the surface, which is the weight of the boy and sled, given as 480 N
So, the friction force can be calculated as:
friction force = 0.060 × 480 N = 28.8 N
Now, we can use the equation:
friction force = mass × acceleration
Since the mass is not given, we can use the weight and the acceleration due to gravity to find it:
weight = mass × acceleration due to gravity
Rearranging the equation, we find:
mass = weight / acceleration due to gravity
Substituting the values, we get:
mass = 480 N / 9.8 m/s^2 ≈ 49 kg
Now, we can find the acceleration using the friction force equation:
28.8 N = 49 kg × acceleration
Rearranging the equation, we find:
acceleration = 28.8 N / 49 kg ≈ 0.59 m/s^2
We can use another equation to find the distance traveled before coming to rest:
final velocity^2 = initial velocity^2 + 2 × acceleration × distance
Since the final velocity is 0 m/s (since the sled comes to rest), the equation becomes:
0^2 = (6.2 m/s)^2 + 2 × 0.59 m/s^2 × distance
Rearranging the equation, we find:
distance = (0 - (6.2 m/s)^2) / (2 × 0.59 m/s^2)
Simplifying the equation, we get:
distance ≈ -20.63 m^2 / 1.18 m/s^2 ≈ -17.5 m
Since distance cannot be negative, we take the absolute value:
distance ≈ 17.5 m
Therefore, the sled travels approximately 17.5 meters on the level surface before coming to rest.
To find the distance the sled travels on the level surface before coming to rest, we can use the concept of work-energy principle. According to this principle, the total work done on an object is equal to the change in its kinetic energy.
The work done on the sled is given by the product of the force of friction and the distance traveled:
Work = Force × Distance
The force of friction can be calculated using the equation:
Force of friction = coefficient of friction × normal force
The normal force is the force exerted by the ground on the sled and is equal to the weight of the sled (since the sled is on a level surface).
Normal force = weight of the sled = mass × acceleration due to gravity
The mass of the sled can be calculated using the weight formula:
Weight = mass × acceleration due to gravity
Rearranging the formula, we have:
mass = weight / acceleration due to gravity
Now, we can calculate the distance traveled by dividing the work done on the sled by the force of friction:
Distance = Work / Force of friction
To find the work done on the sled, we can use the equation:
Work = change in kinetic energy
The change in kinetic energy is calculated by subtracting the final kinetic energy from the initial kinetic energy:
change in kinetic energy = final kinetic energy - initial kinetic energy
The initial kinetic energy of the sled can be calculated using the formula:
Initial kinetic energy = 0.5 × mass × initial velocity^2
The final kinetic energy is zero because the sled comes to rest.
Now let's calculate the distance traveled by the sled:
Step 1: Calculate the mass of the sled:
mass = weight / acceleration due to gravity
mass = 480 N / 9.8 m/s^2
Step 2: Calculate the initial kinetic energy:
Initial kinetic energy = 0.5 × mass × initial velocity^2
Initial kinetic energy = 0.5 × mass × (6.2 m/s)^2
Step 3: Calculate the work done on the sled:
change in kinetic energy = final kinetic energy - initial kinetic energy
Work = change in kinetic energy = 0 - initial kinetic energy
Step 4: Calculate the force of friction:
Force of friction = coefficient of friction × normal force
Force of friction = coefficient of friction × weight of the sled
Step 5: Calculate the distance traveled by the sled:
Distance = Work / Force of friction
By substituting the values into the above equations, we can find the answer.