A girl coasts down a hill on a sled, reaching
a level surface at the bottom with a speed
of 6.2 m/s. The coefficient of kinetic friction
between runners and snow is 0.059, and the
girl and sled together weigh 736 N.
The acceleration of gravity is 9.8 m/s^2
How far does the sled travel on the level
surface before coming to a rest?
Answer in units of m
vf^2=vi^2+2ad
but a=force/mass= -mu*mg/m=-mu*g the negative sign means opposite to motion.
vf^2=6.2^2-2*mu*g*distance
solve for distance.
To find the distance the sled travels on the level surface before coming to a rest, we can use the equation for the force of kinetic friction:
F_kinetic_friction = coefficient_of_kinetic_friction * normal_force
First, let's find the normal force acting on the sled. The normal force is equal to the weight of the girl and sled:
normal_force = weight_of_girl_and_sled
Given that the weight of the girl and sled is 736 N, the normal force is also 736 N.
Now, we can calculate the force of kinetic friction:
F_kinetic_friction = 0.059 * 736
F_kinetic_friction = 43.424 N
The force of kinetic friction acts opposite to the direction of motion, so it will cause a deceleration. We can use the equation:
F_net = mass * acceleration
Since force is equal to mass times acceleration, we can rearrange the equation to solve for acceleration:
acceleration = F_net / mass
In this case, the net force is the force of kinetic friction and the mass is the combined weight of the girl and sled divided by the acceleration due to gravity. So:
acceleration = F_kinetic_friction / (weight_of_girl_and_sled / acceleration_due_to_gravity)
acceleration = 43.424 / (736 / 9.8)
acceleration = 0.437 m/s^2
Now, we can use the equation of motion to find the distance the sled travels on the level surface:
v^2 = u^2 + 2as
Where:
v = final velocity (0 m/s in this case)
u = initial velocity (6.2 m/s)
a = acceleration (-0.437 m/s^2)
s = distance
Rearranging the equation, we have:
s = (v^2 - u^2) / (2a)
s = (0 - (6.2)^2) / (2 * -0.437)
s = (-38.44) / (-0.874)
s = 44 m
Therefore, the sled travels a distance of 44 meters on the level surface before coming to a rest.
To find the distance the sled travels on the level surface before coming to rest, we need to determine the deceleration caused by the friction forces acting on the sled.
First, we can calculate the friction force using the equation:
Friction force = coefficient of kinetic friction × weight of the sled
Given that the coefficient of kinetic friction is 0.059 and the weight of the sled is 736 N, we can calculate:
Friction force = 0.059 × 736 N
Next, we can find the deceleration caused by the friction force using Newton's second law of motion:
Force = mass × acceleration
Since the sled's mass is not given but the weight is, we can use the equation:
Weight = mass × acceleration due to gravity
Rearranging the equation, we can find the mass of the sled:
Mass = weight / acceleration due to gravity
Given that the acceleration due to gravity is 9.8 m/s^2, we can calculate:
Mass = 736 N / 9.8 m/s^2
Now that we have the mass, we can find the deceleration using Newton's second law:
Friction force = mass × deceleration
Rearranging the equation, we can find the deceleration:
Deceleration = friction force / mass
Now, we can calculate:
Deceleration = (0.059 × 736 N) / (736 N / 9.8 m/s^2)
With the deceleration calculated, we can use the kinematic equation to find the distance traveled by the sled:
v^2 = u^2 + 2as
where:
v = final velocity (0 m/s, as the sled comes to rest)
u = initial velocity (6.2 m/s)
a = deceleration (calculated above)
s = distance traveled (what we're solving for)
Rearranging the equation gives us:
s = (v^2 - u^2) / (2a)
Substituting the given values, we have:
s = (0^2 - 6.2^2) / (2 × deceleration)
Finally, we can substitute the calculated deceleration value and solve for s:
s = (0^2 - 6.2^2) / (2 × calculated deceleration)
After the calculation, we will have the distance traveled by the sled on the level surface before coming to a rest.