Final answer:
The speed of the block is 4.08 m/s, the magnitude of the block’s acceleration is 25.41 m/s^2, and the direction of the block’s acceleration is toward the wall.
Step-by-step explanation:
(a) To find the speed of the block, we can use the principle of conservation of mechanical energy. The potential energy stored in the spring when it is compressed is converted into the kinetic energy of the block when it is released. The potential energy stored in the spring is given by:
PE = 0.5 * k * x^2
where k is the force constant of the spring and x is the compression of the spring. Plugging in the values, we get:
PE = 0.5 * 130.0 N/m * 0.80 m * 0.80 m = 41.60 J
The kinetic energy of the block when it is released is given by:
KE = 0.5 * m * v^2
where m is the mass of the block and v is its speed. Equating the potential and kinetic energies, we have:
PE = KE
41.60 J = 0.5 * 3.00 kg * v^2
Solving for v, we get:
v = √(41.60 J / (0.5 * 3.00 kg)) = 4.08 m/s
(b) The magnitude of the block's acceleration can be calculated using Newton's second law, which states that the net force acting on an object is equal to its mass multiplied by its acceleration. In this case, the net force is the force applied to the block minus the force of friction. The force applied to the block is given by F = 88.0 N. The force of friction can be calculated using the equation:
f = μk * m * g
where μk is the coefficient of kinetic friction, m is the mass of the block, and g is the acceleration due to gravity. Plugging in the values, we get:
f = 0.400 * 3.00 kg * 9.8 m/s^2 = 11.76 N
The net force is therefore:
net force = F - f = 88.0 N - 11.76 N = 76.24 N
Using Newton's second law, we have:
76.24 N = 3.00 kg * a
Solving for a, we get:
a = 76.24 N / 3.00 kg = 25.41 m/s^2
(c) The direction of the block's acceleration can be determined by considering the net force acting on the block. In this case, the applied force and the force of friction are in opposite directions, resulting in a net force in the direction of the applied force. Therefore, the direction of the block's acceleration is toward the wall.