A 58.4-kg person, running horizontally with a velocity of +3.84 m/s, jumps onto a 10.7-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. (Indicate the direction with the sign of your answer.)(b) The sled and person coast 31.5 m on level snow before coming to rest. What is the coefficient of kinetic friction between the sled and the snow?
(a) To find the velocity of the sled and person as they move away, we can use the principle of conservation of momentum. The total momentum before the collision must equal the total momentum after the collision:
m1 * v1 + m2 * v2 = (m1 + m2) * vf
where m1 and v1 are the mass and velocity of the person, m2 and v2 are the mass and velocity of the sled, and vf is the final velocity of the combined person and sled.
Since the sled is initially at rest, v2 = 0, and the equation simplifies to:
m1 * v1 = (m1 + m2) * vf
Now we can plug in the given values and solve for vf:
(58.4 kg) * (+3.84 m/s) = (58.4 kg + 10.7 kg) * vf
vf = (58.4 kg * 3.84 m/s) / (58.4 kg + 10.7 kg) = 224.256 / 69.1 = 3.244 m/s
So the velocity of the sled and person as they move away is +3.244 m/s, with the positive sign indicating that they move in the same direction as the person was initially running.
(b) To find the coefficient of kinetic friction between the sled and the snow, we can apply Newton's second law:
f_friction = μ * f_norm
where f_friction is the frictional force, μ is the coefficient of kinetic friction, and f_norm is the normal force acting on the sled and person. Since there is no vertical acceleration, the normal force equals the combined weight of the person and sled:
f_norm = (m1 + m2) * g = 69.1 kg * 9.81 m/s^2 = 677.031 N
The work-energy principle states that the work done by friction (W_friction) must equal the change in kinetic energy:
W_friction = μ * f_norm * d
where d is the distance traveled before coming to rest (31.5 m). The change in kinetic energy is given by:
ΔKE = 0.5 * (m1 + m2) * vf^2 - 0.5 * (m1 + m2) * 0^2
Since vf = 3.244 m/s:
ΔKE = 0.5 * 69.1 kg * (3.244 m/s)^2 = 365.438 J
Now we can set up an equation using the work-energy principle:
μ * f_norm * d = ΔKE
μ * 677.031 N * 31.5 m = 365.438 J
μ = 365.438 J / (677.031 N * 31.5 m) = 0.0166
So the coefficient of kinetic friction between the sled and the snow is 0.0166.
To solve this problem, we can use the principle of conservation of momentum.
(a)
The principle of conservation of momentum states that the total momentum before a collision is equal to the total momentum after the collision, provided no external forces act on the system.
Before the collision:
Person's mass (m1) = 58.4 kg
Person's velocity (v1) = +3.84 m/s
Sled's mass (m2) = 10.7 kg
Sled's initial velocity (v2) = 0 m/s (at rest)
Total initial momentum (before collision) = m1 * v1 + m2 * v2
After the collision:
Let the final velocity of the person and sled be v.
Total final momentum (after collision) = (m1 + m2) * v
According to the principle of conservation of momentum:
m1 * v1 + m2 * v2 = (m1 + m2) * v
Plugging in the given values:
(58.4 kg) * (3.84 m/s) + (10.7 kg) * (0 m/s) = (58.4 kg + 10.7 kg) * v
Simplifying the equation:
(58.4 kg * 3.84 m/s) = (69.1 kg) * v
v = (58.4 kg * 3.84 m/s) / (69.1 kg)
v = 3.242 m/s (approximately)
Therefore, the velocity of the sled and person as they move away is approximately +3.242 m/s. The positive sign indicates that the direction of their motion is in the same direction as the initial velocity of the person.
(b)
To find the coefficient of kinetic friction between the sled and the snow, we need to consider the work done by friction.
The work done by friction can be calculated using the equation:
Work = Force * Distance * cos(theta)
In this case, the work done by friction will be equal to the change in kinetic energy of the person and the sled.
The change in kinetic energy can be calculated using the equation:
Change in kinetic energy = (1/2) * Mass * (Final Velocity)^2 - (1/2) * Mass * (Initial Velocity)^2
In this case, the initial velocity of the sled and person is v (3.242 m/s), and the final velocity is 0 m/s (because they come to rest).
The distance traveled by the sled and person before coming to rest is given as 31.5 m.
Therefore, the equation for the change in kinetic energy becomes:
Change in kinetic energy = (1/2) * (m1 + m2) * (0 m/s)^2 - (1/2) * (m1 + m2) * (3.242 m/s)^2
Next, we can find the work done by friction:
Work = Change in kinetic energy
Using the equation for work done by friction:
Work = Force * Distance * cos(theta)
In this case, the force is the force of kinetic friction (Fk), the distance is 31.5 m, and theta is the angle between the force and displacement (which is 0 degrees for horizontal motion).
Therefore:
Fk * 31.5 m * cos(0 degrees) = - Change in kinetic energy
Since the sled and person come to rest and the final kinetic energy is 0, the equation becomes:
Fk * 31.5 m * cos(0 degrees) = 0
Therefore, the force of kinetic friction is 0.
The coefficient of kinetic friction (μk) can be calculated using the equation:
μk = Fk / (m1 + m2) * g
where g is the acceleration due to gravity.
Since the force of kinetic friction is 0, the coefficient of kinetic friction is also 0.
Therefore, the coefficient of kinetic friction between the sled and the snow is 0.
To solve this problem, we can use the principles of conservation of momentum to analyze the collision between the person and the sled. Then, we can use the principles of work and energy to find the coefficient of kinetic friction between the sled and the snow.
(a) To find the velocity of the sled and person as they move away after the collision, we can use the conservation of momentum equation:
m1 * v1 + m2 * v2 = (m1 + m2) * V
Where:
m1 is the mass of the person
v1 is the initial velocity of the person
m2 is the mass of the sled
v2 is the initial velocity of the sled
V is the final velocity of both the person and the sled
From the given information:
m1 = 58.4 kg
v1 = 3.84 m/s
m2 = 10.7 kg
v2 = 0 since the sled is initially at rest
Plugging the values into the equation, we have:
58.4 kg * 3.84 m/s + 10.7 kg * 0 = (58.4 kg + 10.7 kg) * V
Calculating:
224.256 kg∙m/s = 69.1 kg * V
V = 224.256 kg∙m/s / 69.1 kg ≈ 3.244 m/s
Therefore, the velocity of the sled and person as they move away after the collision is approximately 3.244 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 is equal to the change in kinetic energy of the sled and the person.
Since the sled and the person eventually come to rest, the work done by friction is equal to the initial kinetic energy:
Work = ΔKE
The initial kinetic energy can be calculated as:
KE_initial = (1/2) * m_total * V^2
Where:
m_total is the total mass of the sled and the person, which is equal to m1 + m2
V is the final velocity of the sled and the person, which is 3.244 m/s (from part (a))
From the given values:
m1 = 58.4 kg
m2 = 10.7 kg
m_total = m1 + m2 ≈ 58.4 kg + 10.7 kg ≈ 69.1 kg
Plugging the values into the equation, we have:
KE_initial = (1/2) * 69.1 kg * (3.244 m/s)^2
KE_initial ≈ 353.122 joules
Since the work done by friction is equal to the initial kinetic energy, we can write:
Work = μ * m_total * g * d
Where:
μ is the coefficient of kinetic friction we want to find
m_total is the total mass, which is 69.1 kg
g is the acceleration due to gravity, approximately 9.8 m/s^2
d is the distance traveled, which is given as 31.5 m
Plugging the values into the equation, we have:
353.122 joules = μ * 69.1 kg * 9.8 m/s^2 * 31.5 m
Simplifying:
353.122 joules = μ * 21527.46 kg∙m^2/s^2
Now, we can solve for μ:
μ ≈ 353.122 joules / (21527.46 kg∙m^2/s^2) ≈ 0.0164
Therefore, the coefficient of kinetic friction between the sled and the snow is approximately 0.0164.