A small girl and boy take a sled up to the top of a snow covered slope which is inclined 30 degrees above the horizontal. The girl, boy and the sled have a mass of 160 kg. The coefficient of static friction is 0.28 and the coefficient of kinetic friction is 0.14 between the sled and the snow. They hop on, push off and slide down the 30 meter long incline.

a. How fast are they going at the bottom?
b. How long does it take them to go the 30 meters

μ=0.14, s=30

m=160 kg

x: m•a= m•g•sinα - F(fr),
y: 0= - m•g•cosα +N.

F(fr)=μ•N= μ• m•g•cosα
m•a= m•g•sinα - μ• m•g•cosα
a= g(sinα - μ•cosα)
v=sqrt(2as)
t=v/a

To calculate the speed at the bottom of the slope and the time taken to slide down, we need to break down the problem into different steps. Let's calculate them one by one:

Step 1: Find the force of gravity acting on the sled and children.
The force of gravity can be calculated using the formula: F_gravity = mass * gravitational acceleration.
F_gravity = 160 kg * 9.8 m/s^2 (acceleration due to gravity) = 1568 N.

Step 2: Find the normal force.
The normal force is the force exerted by a surface to support the weight of an object resting on it. In this case, the normal force acts perpendicular to the slope. It can be calculated using the formula: N = mg * cos(theta).
N = (160 kg * 9.8 m/s^2) * cos(30 degrees) = 1320 N.

Step 3: Find the force of friction.
The first type of friction we encounter is static friction, which is present until the object starts sliding. The formula to calculate static friction is: F_friction_static = coefficient of static friction * N.
F_friction_static = 0.28 * 1320 N = 369.6 N.

Now, we need to determine if the static friction is greater than the force applied. If it is greater, the sled will not slide. If it is less, the sled will begin to slide.

Step 4: Determine if the sled slides initially.
The force applied horizontally is the force the children exert to push off. In this case, there are two children; therefore, the force applied is doubled.
Force_applied = 2 * F_gravity * sin(theta).
Force_applied = 2 * (160 kg * 9.8 m/s^2) * sin(30 degrees) = 627.2 N.

Since the force applied (627.2 N) is less than the static friction force (369.6 N), the sled starts to slide.

Step 5: Find the acceleration on the slope.
The net force acting on the sled and children on the incline is the difference between the force applied and the force of kinetic friction.
Net_force = Force_applied - F_friction_kinetic.
Net_force = Force_applied - (coefficient of kinetic friction * N).
Net_force = 627.2 N - (0.14 * 1320 N) = 448.8 N.

Now that we have the net force acting on the sled, we can calculate the acceleration using Newton's second law: F_net = mass * acceleration.
448.8 N = 160 kg * acceleration.
acceleration = 2.805 m/s^2.

Step 6: Calculate the final speed at the bottom.
To calculate the speed at the bottom of the slope, we will use the kinematic equation: v^2 = u^2 + 2as, where u is the initial velocity (0 m/s), a is the acceleration (2.805 m/s^2), and s is the distance traveled (30 m).
v^2 = (0 m/s)^2 + 2 * 2.805 m/s^2 * 30 m.
v^2 = 168.3 m^2/s^2.
v = sqrt(168.3 m^2/s^2) ≈ 12.99 m/s.

Step 7: Calculate the time taken to slide down.
To find the time taken to slide down the 30-meter incline, we will use the equation: t = (v - u) / a, where v is the final velocity (12.99 m/s), u is the initial velocity (0 m/s), and a is the acceleration (2.805 m/s^2).
t = (12.99 m/s - 0 m/s) / 2.805 m/s^2.
t ≈ 4.63 s.

Therefore,
a. The speed at the bottom is approximately 12.99 m/s.
b. The time taken to slide down the 30-meter incline is approximately 4.63 seconds.