A 60.5-kg person, running horizontally with a velocity of +3.76 m/s, jumps onto a 10.4-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.
m/s
(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?
To solve this problem, we can use the principle of conservation of momentum.
(a)
The momentum before the collision is equal to the momentum after the collision. The momentum of an object is given by the product of its mass and velocity.
Before the collision:
Person's momentum = mass of person * velocity of person
= 60.5 kg * 3.76 m/s
= 227.38 kg·m/s
Sled's momentum = mass of sled * velocity of sled
= 10.4 kg * 0 m/s (initially at rest)
= 0 kg·m/s
Total momentum before the collision = momentum of person + momentum of sled
= 227.38 kg·m/s + 0 kg·m/s
= 227.38 kg·m/s
After the collision, both the person and the sled will move together as one system. Let's assume their velocity after the collision is V.
Total momentum after the collision = (mass of person + mass of sled) * velocity after collision
= (60.5 kg + 10.4 kg) * V
= 70.9 kg * V
According to the principle of conservation of momentum:
Total momentum before collision = Total momentum after collision
227.38 kg·m/s = 70.9 kg * V
Dividing both sides of the equation by 70.9 kg:
V = 227.38 kg·m/s ÷ 70.9 kg
V ≈ 3.2 m/s
Therefore, the velocity of the sled and person as they move away is approximately 3.2 m/s.
(b)
To find the coefficient of kinetic friction between the sled and the snow, we can use the equations of motion.
The work done by the friction force is equal to the change in kinetic energy of the sled and the person. We can express this mathematically as:
Work (W) = Change in kinetic energy (ΔK)
The work done by the friction force is given by the product of the force of friction and the distance over which it acts. In this case, the work done by the friction force is equal to the force of friction multiplied by the distance coasted by the sled and the person (30.0 m).
Since the sled and the person come to rest, the change in kinetic energy is equal to zero. Hence, W = 0.
The force of friction (F) can be expressed using the coefficient of kinetic friction (μk) and the normal force (N):
F = μk * N
The normal force (N) can be calculated by multiplying the mass (m) by the acceleration due to gravity (g):
N = m * g
Substituting the expression for the normal force into the equation for the force of friction:
F = μk * m * g
Using Newton's second law (F = m * a), we can rewrite this equation as:
μk * m * g = μk * m * a
However, since the sled and person come to rest, the acceleration (a) is equal to zero, so the equation becomes:
μk * m * g = 0
Simplifying further, we can cancel out the mass (m):
μk * g = 0
Dividing both sides of the equation by g:
μk = 0
Therefore, the coefficient of kinetic friction between the sled and the snow is zero.
To solve this problem, we can apply the principle of conservation of momentum.
(a) The initial momentum of the person is given by the product of their mass (m1 = 60.5 kg) and velocity (v1 = 3.76 m/s):
p1 = m1 * v1
The initial momentum of the sled is zero since it is initially at rest:
p2 = 0
After the collision, the person and the sled move together as one system, so their final momentum can be calculated as:
p3 = (m1 + m2) * v
According to the law of conservation of momentum, the initial momentum of the system is equal to the final momentum:
p1 + p2 = p3
Substituting the values:
m1 * v1 + 0 = (m1 + m2) * v
Rearranging the equation to solve for v:
v = (m1 * v1) / (m1 + m2)
Plugging in the given values:
v = (60.5 kg * 3.76 m/s) / (60.5 kg + 10.4 kg)
Calculating the values:
v ≈ 2.35 m/s
Therefore, the velocity of the sled and person as they move away is approximately 2.35 m/s.
(b) To find the coefficient of kinetic friction (μ), we can use the equation:
μ = F_friction / (m1 + m2) * g
Where F_friction is the force of friction, (m1 + m2) is the total mass of the system, and g is the acceleration due to gravity.
To find F_friction, we can use the equation of motion:
F_friction = m1 * a
Where a is the deceleration of the system, calculated using the distance traveled and time taken to come to rest:
a = (v^2 - u^2)/(2 * s)
Where v is the final velocity (0 m/s), u is the initial velocity (2.35 m/s), and s is the distance traveled (30.0 m).
Plugging in the values:
a = (0 - 2.35^2) / (2 * 30.0)
Calculating the values:
a = -0.0911 m/s^2
Now, substituting the values of m1, m2, and g, we can calculate the coefficient of kinetic friction:
μ = (m1 * a) / ((m1 + m2) * g)
Plugging in the values:
μ = (60.5 kg * -0.0911 m/s^2) / ((60.5 kg + 10.4 kg) * 9.8 m/s^2)
Calculating the values:
μ ≈ -0.018
Note: The negative sign indicates the direction of the frictional force, opposing the motion.
Therefore, the coefficient of kinetic friction between the sled and the snow is approximately -0.018.
This one I'll help you with
a)60.5+10.4=70.9
70.9/3.76=18.856382
b)18.856382/30.0=0.628546