Final answer:
Using the work-energy theorem, we can determine the velocity of the block at the top of the incline by calculating the work done by gravity and friction and the initial and final kinetic energies of the block. The final velocity of the block is approximately 10.5 m/s.
Step-by-step explanation:
To determine the velocity of the block at the top of the incline, we can use the work-energy theorem. The work-energy theorem states that the work done on an object is equal to the change in its kinetic energy. In this case, the work done on the block is equal to the force of gravity multiplied by the distance moved along the incline, minus the work done by friction.
First, let's calculate the work done by gravity. The force of gravity can be calculated using the mass of the block and the acceleration due to gravity (9.8 m/s^2): Fg = m * g = 10 kg * 9.8 m/s^2 = 98 N. The work done by gravity is equal to the force of gravity multiplied by the distance moved along the incline: Wg = Fg * d = 98 N * 10 m = 980 J.
Next, let's calculate the work done by friction. Since the force of friction is constant and equal to 100 N, and the distance moved along the incline is 10 m, the work done by friction is equal to the force of friction multiplied by the distance moved along the incline: Wf = Ff * d = 100 N * 10 m = 1000 J.
Now, let's calculate the initial kinetic energy of the block. The initial velocity of the block is given as 15 m/s, and the mass of the block is 10 kg. The initial kinetic energy is given by the formula: KEi = 0.5 * m * v^2 = 0.5 * 10 kg * (15 m/s)^2 = 1125 J.
Using the work-energy theorem, we can calculate the final kinetic energy of the block: Wf - Wg = KEf - KEi. Rearranging the formula, we get: KEf = KEi + Wg - Wf = 1125 J + 980 J - 1000 J = 1105 J. Finally, we can calculate the final velocity of the block using the final kinetic energy: KEf = 0.5 * m * v^2. Rearranging the formula, we get: v = sqrt((2 * KEf) / m) = sqrt((2 * 1105 J) / 10 kg) ≈ 10.5 m/s.