A 60.0 kg person, running horizontally with a velocity of 3.80 m/s, jumps onto a 12.0 kg
sled that is initially at rest. a) Ignoring the effects of friction during the collision, find the
velocity of the sled and person as they move away. b) The sled and person coast 30.0 m
on level snow before coming to rest. What is the coefficient of kinetic friction between
the sled and the snow?
Knock, knock.
Who's there?
Sledding physics.
Sledding physics who?
Sledding physics is about to make you go "whoa!"
Now, let's get serious about answering your question.
a) To find the velocity of the sled and person after the collision, we can apply the law of conservation of momentum. The initial momentum is given by the product of the person's mass and velocity, plus the sled's initial velocity (which is zero):
Initial momentum = (60.0 kg)(3.80 m/s) + (12.0 kg)(0 m/s) = 228 kg*m/s
Since momentum is conserved, the final momentum should also be 228 kg*m/s. We can assume that the person and sled stick together after the collision, so their masses will add up to 60.0 kg + 12.0 kg = 72.0 kg. Now we can rearrange the equation to find their final velocity:
Final velocity = (Final momentum) / (Total mass)
Final velocity = 228 kg*m/s / 72.0 kg = 3.17 m/s
So, the sled and person move away with a velocity of 3.17 m/s after the collision.
b) To find the coefficient of kinetic friction between the sled and the snow, we can use the equation for work done by friction:
Work done by friction = Force of friction * Distance
The force of friction can be calculated using the equation:
Force of friction = coefficient of kinetic friction * Normal force
In this case, the normal force equals the weight of the sled and person, which is the mass multiplied by the acceleration due to gravity:
Normal force = (60.0 kg + 12.0 kg) * 9.8 m/s^2 = 720 N
The work done by friction is equal to the initial kinetic energy of the sled and person:
Work done by friction = (1/2) * (Total mass) * (Initial velocity)^2
Now, let's plug in the numbers:
Work done by friction = (1/2) * (72.0 kg) * (3.17 m/s)^2 = 359.72 J
Finally, since the work done by friction is also equal to the product of the force of friction and the distance, we can rearrange the equation to find the coefficient of kinetic friction:
coefficient of kinetic friction = (Work done by friction) / (Force of friction * Distance)
coefficient of kinetic friction = 359.72 J / (720 N * 30.0 m) = 0.167
So, the coefficient of kinetic friction between the sled and the snow is approximately 0.167.
Remember, it's not all fun and games when it comes to physics, but we can still find some humor in it!
To solve this problem, we can use the law of conservation of momentum and the laws of motion.
a) First, let's find the initial momentum and final momentum of the system.
The initial momentum is equal to the mass of the person multiplied by their initial velocity:
Initial momentum (person) = mass(person) x velocity(person)
= 60.0 kg x 3.80 m/s
= 228 kg·m/s
The initial momentum of the sled is zero since it is initially at rest:
Initial momentum (sled) = 0 kg·m/s
The final momentum of the system is the sum of the momentum of the person and the sled:
Final momentum (system) = Final momentum (person) + Final momentum (sled)
We can assume that the momentum is conserved during the collision, so the final momentum of the system will be the same as the initial momentum:
Final momentum (system) = 228 kg·m/s
Since the person jumps onto the sled, they continue moving together as a system. Let's call the final velocity of the system as "v":
Final momentum (system) = (mass(person) + mass(sled)) x v
= (60.0 kg + 12.0 kg) x v
= 72.0 kg x v
Equating the final momentum with the initial momentum, we get:
72.0 kg x v = 228 kg·m/s
Solving for "v":
v = 228 kg·m/s / 72.0 kg
v ≈ 3.17 m/s
Therefore, the velocity of the sled and person as they move away is approximately 3.17 m/s in the forward direction.
b) Now, let's find the coefficient of kinetic friction between the sled and the snow.
The work done by friction is given by:
Work (friction) = Force (friction) x distance
The work done by friction will be equal to the change in kinetic energy of the system:
Work (friction) = Change in KE (system)
The change in KE (system) is given by:
Change in KE (system) = KE (initial) - KE (final)
Since the sled and person come to rest, the final kinetic energy is zero:
Change in KE (system) = KE (initial) - 0
= KE (initial)
The initial kinetic energy is given by:
KE (initial) = (1/2) x (mass(person) + mass(sled)) x (velocity(person) + velocity(sled))^2
Plugging in the given values:
KE (initial) = (1/2) x (60.0 kg + 12.0 kg) x (3.80 m/s + 3.17 m/s)^2
≈ 747.18 J
Therefore, the work done by friction will be 747.18 J.
Since the force of friction is opposite to the direction of motion, it can be calculated as:
Force (friction) = - Work (friction) / distance
Plugging in the given distance of 30.0 m:
Force (friction) = - 747.18 J / 30.0 m
≈ - 24.91 N
In order to calculate the coefficient of friction, we need to find the normal force acting on the sled. Since the sled and person are on level snow, the normal force is equal to the weight of the system:
Normal force = Weight (system)
= (mass(person) + mass(sled)) x gravitational acceleration
Plugging in the given values:
Normal force = (60.0 kg + 12.0 kg) x 9.8 m/s^2
= 72.0 kg x 9.8 m/s^2
= 705.6 N
The force of friction can be calculated using the following equation:
Force (friction) = coefficient of kinetic friction x Normal force
Plugging in the known values and solving for the coefficient of kinetic friction:
- 24.91 N = coefficient of kinetic friction x 705.6 N
coefficient of kinetic friction ≈ - 24.91 N / 705.6 N
≈ - 0.035
Since the coefficient of friction cannot be negative, we take the absolute value:
coefficient of kinetic friction ≈ 0.035
Therefore, the coefficient of kinetic friction between the sled and the snow is approximately 0.035.
To find the answers to these questions, we need to apply the principles of conservation of momentum and work done by friction.
a) First, let's calculate the initial momentum of the person before jumping onto the sled. The formula for momentum is given by:
Momentum = mass * velocity
The mass of the person is 60.0 kg, and the velocity is 3.80 m/s. So, the initial momentum of the person is:
Momentum_person = 60.0 kg * 3.80 m/s = 228 kg m/s
Since we are assuming no external forces (like friction) are acting on the system during the collision, the total momentum before and after the collision should be conserved.
The sled is initially at rest, so its initial momentum is zero:
Momentum_sled_initial = 0 kg m/s
Now, when the person jumps onto the sled, both the person and the sled move together. Let's assume their final velocity as v (the velocity of the sled and person as they move away).
The final momentum of the person and the sled together is:
Momentum_person+sled_final = (mass_person + mass_sled) * v
Where mass_person is 60.0 kg and mass_sled is 12.0 kg.
Using the principle of conservation of momentum, we can equate the initial and final momentum together:
Momentum_person + Momentum_sled_initial = Momentum_person+sled_final
228 kg m/s + 0 kg m/s = (60.0 kg + 12.0 kg) * v
Simplifying the equation:
228 kg m/s = 72.0 kg * v
Dividing both sides by 72.0 kg:
v = 228 kg m/s / 72.0 kg
v ≈ 3.17 m/s
Therefore, the velocity of the sled and person as they move away is approximately 3.17 m/s.
b) To find the coefficient of kinetic friction between the sled and the snow, we can use the work-energy principle.
The work done by friction can be calculated using the formula:
Work_friction = Force_friction * distance
The work done by friction is equal to the kinetic energy dissipated during the sled's motion.
The kinetic energy is given by the formula:
Kinetic energy = 0.5 * mass * velocity^2
Given that the sled and person coast 30.0 m on level snow before coming to rest, we can calculate the kinetic energy using their final velocity (as calculated in part a)) and the mass of the system (the person and the sled):
Kinetic energy = 0.5 * (mass_person + mass_sled) * velocity^2
Kinetic energy = 0.5 * (60.0 kg + 12.0 kg) * (3.17 m/s)^2
Kinetic energy ≈ 630.6 J
The work done by friction is equal to the kinetic energy:
Work_friction = 630.6 J
Since the force of friction is in the opposite direction of motion, we can express it as:
Force_friction = - work_friction / distance
Given that the distance is 30.0 m, we can substitute the values into the equation:
Force_friction = -630.6 J / 30.0 m
Force_friction = -21.02 N
The force of friction can be represented as the product of the coefficient of kinetic friction and the normal force between the sled and the snow. The normal force is equal to the weight of the sled (mass_sled * gravity).
Force_friction = coefficient_of_friction * (mass_sled * gravity)
Substituting the values:
-21.02 N = coefficient_of_friction * (12.0 kg * 9.8 m/s^2)
Dividing both sides by (12.0 kg * 9.8 m/s^2):
coefficient_of_friction = -21.02 N / (12.0 kg * 9.8 m/s^2)
coefficient_of_friction ≈ -0.18
Since the coefficient of friction cannot be negative, we discard the negative sign and take the absolute value:
coefficient_of_friction ≈ 0.18
Therefore, the coefficient of kinetic friction between the sled and the snow is approximately 0.18.