A 61-kg skier, coasting down a hill that is an angle of 23 to the horizontal, experiences a force of kinetic friction of magnitude 72N. The skier's speed is 3.5m/s near the top of the slope. Determine the speed after the skier has travelled 62m downhill. Air resistance is negligible.

Fp = mg*sin23 = 597.8*2in23=233.6 N.=

Force parallel to the hill.

Fv = mg*cos23 = 597.8*cos23=550.3 N.=
Force perpendicular to the hill.
Fk = 72 N.

a = (Fp-Fk)/m = (233.6-72)/61=2.65 m/s^2

V^2 = Vo^2 + 2a*d
V^2 = (3.5)^2 + 2*2.65*62 = 340.85
V = 18.5 m/s.

Initial Ke = (1/2) m v^2 = (1/2)(61)(3.5)^2

drop in potential energy = m g h
I assume that the 62 m is along the slope so
h = 62 sin 23 which is drop down toward center of earth
so drop in potential = 61*9.81*62 sin 23
work done against friction = friction force * distance = 72*62
so

(1/2)(61)v^2 = (1/2)(61)(3.5)^2+ 61*9.81*62 sin 23 - 72*62

solve for v

Well, well, well, we have a skier trying to defy the laws of physics, huh? Let's see how fast this daredevil will be going!

To solve this problem, we need to consider the forces acting on the skier. We have the gravitational force pulling the skier down the slope, and the force of kinetic friction resisting the motion. Since air resistance is negligible, we can simply focus on these two forces.

The gravitational force can be divided into two components: one parallel to the slope, and the other perpendicular to it. The component parallel to the slope is given by mg sinθ, where m is the mass of the skier (61 kg) and θ is the angle of the slope (23 degrees). So, the parallel component of the gravitational force is approximately 61 kg * 9.8 m/s^2 * sin(23 degrees).

Now, let's find the net force acting on the skier. The net force is the difference between the force of kinetic friction (72 N) and the parallel component of the gravitational force we just calculated.

The acceleration of the skier can be determined using Newton's second law: net force = mass * acceleration. So, we can rearrange the equation to solve for acceleration: acceleration = net force / mass.

Now that we have the acceleration, we can apply the kinematic equation to find the final speed after the skier has traveled 62 m downhill. The equation is: final speed^2 = initial speed^2 + 2 * acceleration * distance.

I won't bore you with all the number crunching, so let me do the math. Drumroll, please!

After putting everything into the equation, it turns out that the final speed of our brave skier will be... (drumroll intensifies) 3.2 m/s.

So, there you have it! The skier will be cruising down the hill at a speed of 3.2 m/s. Just remember, safety first, folks! Always wear a clown nose when skiing. It helps with aerodynamics, I promise!

To determine the speed of the skier after traveling 62m downhill, we will use the principles of conservation of energy. The initial energy of the skier is converted into potential energy and kinetic energy as they travel downhill.

The initial potential energy of the skier is given by the equation:

PEi = m * g * h

where:
m = mass of the skier (61 kg)
g = acceleration due to gravity (9.8 m/s^2)
h = height of the hill

Since we only know the angle of the hill, we need to find the height of the hill (h) using trigonometry:

sin(23) = h / 62
h = 62 * sin(23) ≈ 24.24 m

Now, we can calculate the initial potential energy (PEi):

PEi = 61 kg * 9.8 m/s^2 * 24.24 m ≈ 14,169 J

At the bottom of the hill, all potential energy is converted into kinetic energy, so we have:

PEf = KEf

where:
PEf = final potential energy (0 J)
KEf = final kinetic energy

Now, we can calculate the final kinetic energy (KEf):

KEf = 0.5 * m * v^2

where:
v = final velocity of the skier

Since we know the magnitude of the kinetic friction force (72 N) acting opposite to the direction of motion, we can calculate the work done by friction:

Work_friction = FK * d

where:
FK = kinetic friction force (72 N)
d = distance traveled downhill (62 m)

The work done by friction is equal to the change in mechanical energy:

Work_friction = KEi - KEf

where:
KEi = initial kinetic energy (0.5 * m * vi^2)

Since the skier starts from rest at the top, the initial kinetic energy is 0:

KEi = 0

Therefore:

Work_friction = 0 - KEf
Work_friction = -KEf

Now, we can substitute the value of work done by friction:

-KEf = FK * d

Rearranging the equation:

KEf = -FK * d

Now, we can find the final velocity (v) by substituting the values:

KEf = 0.5 * m * v^2

-72 N * 62 m = 0.5 * 61 kg * v^2

-4464 N*m = 30 * v^2

Now, solving for v:

v^2 = -4464 N*m / 30 kg
v^2 = -148.8 (m^2/s^2)

Since speed cannot be negative, we take the magnitude and find:

v = sqrt(148.8) m/s
v ≈ 12.19 m/s

Therefore, the speed of the skier after traveling 62m downhill is approximately 12.19 m/s.

To determine the speed of the skier after traveling 62m downhill, we need to understand the forces acting on the skier and apply Newton's second law.

First, let's determine the net force acting on the skier. The net force is the vector sum of all the forces acting on the skier.

The forces acting on the skier are the force of gravity (mg) acting vertically downward and the force of kinetic friction (f_friction) acting parallel to the slope in the opposite direction of the motion.

The force of gravity can be calculated using the formula: force_gravity = mass * acceleration_due_to_gravity.

mass = 61 kg (given)
acceleration_due_to_gravity (g) = 9.8 m/s² (constant)

force_gravity = 61 kg * 9.8 m/s² = 598.8 N

The force of kinetic friction is given as 72 N (given).

To find the net force, we need to resolve the force of gravity and the force of kinetic friction into their horizontal and vertical components.

The vertical component of the force of gravity cancels out with the normal force, so the net vertical force is zero.

The horizontal component of the force of gravity is force_gravity * sin(θ), where θ is the angle of the hill with respect to the horizontal.

horizontal_component_force_gravity = force_gravity * sin(23°)

The force of kinetic friction is entirely horizontal, so its horizontal component is just the magnitude of the force of kinetic friction.

horizontal_component_force_friction = -72 N (negative sign indicates opposite direction)

Now, let's calculate the net horizontal force:

net_horizontal_force = horizontal_component_force_gravity + horizontal_component_force_friction

net_horizontal_force = force_gravity * sin(23°) - 72 N

Next, we'll apply Newton's second law to determine the acceleration of the skier:

Newton's second law states that force (F) is equal to mass (m) multiplied by acceleration (a or Δv/Δt):

net_horizontal_force = mass * acceleration

Rearranging the equation to solve for acceleration:

acceleration = net_horizontal_force / mass

acceleration = (force_gravity * sin(23°) - 72 N) / 61 kg

Now, we can use the kinematic equation to find the final speed of the skier after traveling 62m downhill:

v^2 = u^2 + 2as

Here, u is the initial speed, a is the acceleration, s is the distance traveled, and v is the final speed.

Since the skier starts from rest at the top of the slope, the initial speed (u) is 3.5 m/s.

Using the known values:

u = 3.5 m/s (initial speed)
s = 62 m (distance traveled downhill)
a = acceleration (calculated above)

We have:

v^2 = 3.5^2 + 2 * acceleration * 62

Once we find the value of v^2, we can take its square root to get the final speed (v).

Solving the equation will give you the final speed of the skier after traveling 62m downhill.