A child on a sled (total mass 41 kg) slides down a hill inclined at an angle θ = 22° at a constant speed 3.5 m/s.

(a) How long does it take the child to travel 40 meters down the slope?



(b) What is the frictional force on the sled?




(c) What is the coefficient of friction between the sled and the slope?




(d) The child reaches the bottom of the hill and slides along the horizontal muddy ground to rest. If the coefficient of friction is no different on the horizontal surface, how far does the sled go before coming to rest?

Normal force = m g cos 22

= 41 * 9.81 * cos 22 = 373 N

Force down slope = m g sin 22 = 151 N

time to slide 40 m = 40/3.5

acceleration is zero so friction force = 151 N

373 mu = 151
so
mu = 151/373

To solve this problem, we need to break it down into several steps.

Step 1: Analyze the forces acting on the child-sled system

(a) Determine the gravitational force acting on the sled:
The gravitational force (Fg) is given by the equation Fg = mg, where m is the mass of the sled and g is the acceleration due to gravity (9.8 m/s^2).

Fg = (41 kg)(9.8 m/s^2)
Fg = 401.8 N

(b) Calculate the component of the gravitational force parallel to the slope:
The component of the gravitational force parallel to the slope is Fg_parallel = Fg * sin(θ), where θ is the angle of the slope (22°).

Fg_parallel = (401.8 N) * sin(22°)
Fg_parallel = 148.2 N

(c) Determine the frictional force on the sled:
Since the sled is moving at a constant speed, the frictional force (Ff) must be equal in magnitude and opposite in direction to the parallel component of the gravitational force.

Ff = Fg_parallel
Ff = 148.2 N

Step 2: Calculate the time taken to travel 40 meters down the slope

We can use the equation for constant velocity to find the time taken.

Velocity = Distance/Time
Time = Distance/Velocity

Time = 40 m / 3.5 m/s
Time ≈ 11.43 seconds

Step 3: Determine the coefficient of friction

The coefficient of friction (µ) can be calculated using the equation µ = Ff / Fn, where Fn is the normal force.

(d) Calculate the normal force acting on the sled:
The normal force (Fn) is equal in magnitude and opposite in direction to the perpendicular component of the gravitational force.

Fn = Fg_perpendicular = Fg * cos(θ)

Fn = (401.8 N) * cos(22°)
Fn = 363.6 N

Now we can calculate the coefficient of friction:

µ = Ff / Fn
µ = (148.2 N) / (363.6 N)
µ ≈ 0.407

Step 4: Calculate the distance traveled on the horizontal muddy ground

Since the coefficient of friction is the same on the horizontal surface, we can use the equation for constant deceleration to find the distance traveled before coming to rest.

Final Velocity = 0 (sled comes to rest)
Initial Velocity = 3.5 m/s
Acceleration = -µg (negative sign indicates deceleration)

Using the equation V^2 = U^2 + 2as, where V is the final velocity, U is the initial velocity, a is the acceleration, and s is the distance traveled, we can solve for s:

0 = (3.5 m/s)^2 + 2(-0.407 * 9.8 m/s^2) * s

0 = 12.25 m^2/s^2 - 7.974 m/s^2 * s

7.974 m/s^2 * s = 12.25 m^2/s^2

s ≈ 1.54 meters

Therefore, the sled goes approximately 1.54 meters on the horizontal muddy ground before coming to rest.