A truck is traveling at 12 m/s down a hill when the brakes on all four wheels lock. The hill makes an angle θ = 15° with respect to the horizontal. The coefficient of kinetic friction between the tires and the road is 0.7. How far does the truck skid before coming to a stop?

Calculate the friction force using the kinetic coefficient of friction, Uk = .7. Call the friction force Ff

Ff = M*g*cos15*Uk

Let the distance it travels be X

When Ff*X equals the initial kinetic energy plus any gravitational potential energy decrease, all of the available energy will used up as frictional heating work, and the block stops.

Ff *X = M g*cos15*Uk*X
= (1/2) M V^2 + M g X sin 15

Solve for X

X = (1/2) (V^2/g)/[cos15*Uk + sin 15]

To find the distance that the truck skids before coming to a stop, we can use the equation for the force of friction and the equation for the distance traveled by an object.

The force of friction is given by the equation:
Frictional Force = coefficient of friction * 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, it is equal to the weight of the truck. The weight of the truck is calculated as:
Weight = mass * gravity

where gravity is approximately 9.8 m/s^2.

To find the mass of the truck, we need to consider that the weight will be supported by both the horizontal and vertical directions. The vertical component will balance the weight, while the horizontal component will provide the force of friction. The horizontal component of the weight is given by:
Weight_horizontal = Weight * sin(θ)

Now, we can calculate the frictional force:
Frictional Force = coefficient of friction * Weight_horizontal

Next, we can use Newton's second law to find the acceleration of the truck:
Frictional Force = mass * acceleration

Rearranging this equation, we find the acceleration:
acceleration = Frictional Force / mass

Finally, we can use the kinematic equation to find the distance traveled by the truck:
distance = (initial velocity)^2 / (2 * acceleration)

Substituting the given values into these equations, we can find the solution.