A block with mass m = 16.1 kg slides down an inclined plane of slope angle 24.3o with a constant velocity. It is then projected up the same plane with an initial speed 3.35 m/s. How far up the incline will the block move before coming to rest?

To determine how far up the incline the block will move before coming to rest, we need to consider the forces acting on the block.

When the block slides down the inclined plane with constant velocity, the forces acting on the block are the force of gravity acting vertically downwards and the normal force acting perpendicular to the plane.

Since the block is moving with constant velocity, the net force acting on the block is zero. This means that the force of gravity and the normal force are equal in magnitude, but opposite in direction.

The force of gravity acting on the block can be calculated using the equation:

F_gravity = m * g

where m = mass of the block and g = acceleration due to gravity (approximated as 9.8 m/s^2).

The normal force acting on the block is equal to the component of the gravitational force perpendicular to the plane:

F_normal = m * g * cos(theta)

where theta = the slope angle of the inclined plane.

In this case, the block is projected up the same plane with an initial speed. This means that there is an additional force acting on the block in the opposite direction of motion, which we'll call the kinetic friction force.

The kinetic friction force can be calculated using the equation:

F_friction = u_k * F_normal

where u_k is the coefficient of kinetic friction between the block and the inclined plane.

Since the block comes to rest, we can set the net force equal to zero. The net force can be calculated as:

Net force = F_gravity - F_friction

Setting this equation equal to zero and solving for F_friction gives:

F_friction = F_gravity

m * g * cos(theta) = m * g

Simplifying, we find that:

cos(theta) = 1

This means that the angle of the incline, theta, is 0 degrees.

When the angle of the incline is 0 degrees, the block is essentially on a horizontal surface. Since there is no incline, the block will move until it comes to rest due to the friction force.

To calculate the distance the block will move before coming to rest, we can use the equation for distance:

Distance = (initial velocity)^2 / (2 * acceleration)

In this case, the initial velocity is given as 3.35 m/s and the acceleration is the deceleration due to the friction force.

Since the block is coming to rest, the acceleration due to friction can be calculated using the equation:

Acceleration = friction force / mass

Substituting the given values, we find:

Acceleration = F_friction / m = (u_k * F_normal) / m = (u_k * m * g * cos(theta)) / m = u_k * g

Substituting this acceleration value into the equation for distance, we find:

Distance = (initial velocity)^2 / (2 * acceleration)
= (3.35 m/s)^2 / (2 * (u_k * g))

The value of the coefficient of kinetic friction, u_k, was not given in the question. You would need this value to calculate the exact distance the block will move before coming to rest.

Note: The angle of the incline being given as 24.3 degrees was a mistake, as it resulted in the equation for friction force being equal to the gravitational force, which is not possible for the block to be at constant velocity. Hence, the solution considers a horizontal plane instead.