A 3.26 kg block located on a horizontal floor is pulled by a cord that exerts a force F = 10.4 N at an angle theta = 29.0° above the horizontal. The coefficient of kinetic friction between the block and the floor is 0.10. What is the speed of the block 3.1 s after it starts moving?

To find the speed of the block after 3.1 seconds, we can break down the problem into several steps:

Step 1: Calculate the net force acting on the block.
- Resolve the applied force into its horizontal and vertical components.
- The horizontal component: F_h = F * cos(theta)
- The vertical component: F_v = F * sin(theta)
- Calculate the force of friction: F_friction = coefficient of kinetic friction * normal force
- The normal force is equal to the weight of the block: F_normal = m * g, where m is the mass of the block and g is the acceleration due to gravity.
- The net force is the sum of the horizontal component of the applied force and the force of friction:
- F_net = F_h - F_friction

Step 2: Calculate the acceleration of the block.
- The acceleration of the block is given by Newton's second law: a = F_net / m

Step 3: Calculate the distance traveled by the block after 3.1 seconds.
- Use the equation for displacement with constant acceleration: x = x_0 + v_0 * t + 0.5 * a * t^2
- Here, x_0 is the initial position of the block, v_0 is the initial velocity of the block (which is 0 since it starts from rest), a is the acceleration calculated in step 2, and t is the time.

Step 4: Calculate the final velocity of the block.
- Use the equation for final velocity with constant acceleration: v = v_0 + a * t
- Here, v_0 is 0 since the block starts from rest, and a and t are the calculated values.

Let's calculate the speed of the block after 3.1 seconds using the given values:

Step 1:
- F_h = F * cos(theta) = 10.4 N * cos(29.0°)
- F_v = F * sin(theta) = 10.4 N * sin(29.0°)
- F_normal = m * g = 3.26 kg * 9.8 m/s^2
- F_friction = coefficient of kinetic friction * F_normal = 0.10 * (3.26 kg * 9.8 m/s^2)
- F_net = F_h - F_friction

Step 2:
- a = F_net / m

Step 3:
- x = x_0 + v_0 * t + 0.5 * a * t^2

Step 4:
- v = v_0 + a * t

We can now calculate the values for each step:

Step 1:
- F_h = 10.4 N * cos(29.0°) ≈ 9.274 N
- F_v = 10.4 N * sin(29.0°) ≈ 4.828 N
- F_normal = 3.26 kg * 9.8 m/s^2 ≈ 31.9488 N
- F_friction = 0.10 * (3.26 kg * 9.8 m/s^2) ≈ 3.183 N
- F_net = F_h - F_friction = 9.274 N - 3.183 N ≈ 6.091 N

Step 2:
- a = F_net / m = 6.091 N / 3.26 kg ≈ 1.870 m/s^2

Step 3:
- x = x_0 + v_0 * t + 0.5 * a * t^2
- As the block starts from rest, the initial position (x_0) and initial velocity (v_0) are 0. Therefore, x = 0 + 0 * 3.1 s + 0.5 * 1.870 m/s^2 * (3.1 s)^2
- Simplifying, x = 0.5 * 1.870 m/s^2 * (3.1 s)^2 ≈ 8.624 m

Step 4:
- v = v_0 + a * t
- As the block starts from rest, the initial velocity (v_0) is 0. Therefore, v = 0 + 1.870 m/s^2 * 3.1 s ≈ 5.787 m/s

Therefore, the speed of the block 3.1 seconds after it starts moving is approximately 5.787 m/s.

To find the speed of the block after it starts moving, we need to first determine the net force acting on the block and then use Newton's second law of motion (F = ma) to find the acceleration. Finally, we can use the equation of motion (v = u + at) to find the speed.

Let's break down the problem step by step:

1. Determine the horizontal force applied by the cord:
We can resolve the applied force F into two components: horizontal and vertical. The horizontal component is given by F_horizontal = F * cos(theta). Using the given values, F = 10.4 N and theta = 29.0°, we can calculate F_horizontal as:
F_horizontal = 10.4 N * cos(29.0°).

2. Calculate the frictional force:
The frictional force acting on the block can be determined using the coefficient of kinetic friction (μ_k) and the normal force (N) exerted by the floor, which is equal to the weight of the block (N = mg). The frictional force is given by f_friction = μ_k * N.

3. Find the net force:
The net force acting on the block is the difference between the horizontal force applied by the cord and the frictional force. Net force (F_net) is given by F_net = F_horizontal - f_friction.

4. Calculate the acceleration:
Using Newton's second law, F_net = ma, we can rearrange the equation to solve for acceleration (a). Thus, a = F_net / m.

5. Determine the speed of the block after 3.1 seconds:
Using the equation of motion v = u + at, where v is the final velocity, u is the initial velocity (which is zero in this case, as the block starts from rest), a is the acceleration, and t is the time taken, we can substitute the values to find the speed of the block after 3.1 seconds.

Remember to convert all angles to radians when using trigonometry functions (degrees * pi / 180).

Now you can follow these steps and use the provided values to find the speed of the block after 3.1 seconds.