A 6.00 kg block located on a horizontal frictionless floor is pulled by a cord that exerts a force F=14.1 N at an angle θ= 19.0° above the horizontal, as shown.

What is the speed of the block 3.50 seconds after it starts moving?

To find the speed of the block, we need to use Newton's second law and apply the concepts of work and energy.

Step 1: Resolve Forces
Since the force is applied at an angle, we need to resolve it into its horizontal and vertical components.
The horizontal component is given by: F(horizontal) = F * cos(θ)
The vertical component is given by: F(vertical) = F * sin(θ)

Step 2: Calculate the Acceleration
The net force in the horizontal direction is equal to the mass of the block multiplied by the acceleration because there is no horizontal force acting on the block other than the applied force.
So, F(horizontal) = m * a
Therefore, a = F(horizontal) / m

Step 3: Calculate the Distance Traveled
To find the distance traveled, we can use the equation for uniformly accelerated motion: s = u*t + 0.5 * a * t^2, where u is the initial velocity, t is time, a is acceleration, and s is the distance traveled.
Since the block is initially at rest, the initial velocity u = 0.

Step 4: Find the Final Velocity
The final velocity can be found using the equation for uniformly accelerated motion: v = u + a*t, where v is the final velocity, u is the initial velocity, a is the acceleration, and t is time.
Since the initial velocity u = 0, the equation simplifies to v = a*t.

Step 5: Calculate the Speed
Speed is defined as the magnitude of velocity, which means it does not have a direction. To find the speed, we only need the magnitude of the final velocity found in step 4.

Let's calculate the speed of the block 3.50 seconds after it starts moving using the given values:

Given:
Mass of the block (m) = 6.00 kg
Force (F) = 14.1 N
Angle (θ) = 19.0°
Time (t) = 3.50 seconds

Step 1: Resolve Forces
F(horizontal) = F * cos(θ) = 14.1 * cos(19.0°)
F(vertical) = F * sin(θ) = 14.1 * sin(19.0°)

Step 2: Calculate the Acceleration
a = F(horizontal) / m

Step 3: Calculate the Distance Traveled
s = u*t + 0.5 * a * t^2 (Since the block starts from rest, u = 0)

Step 4: Find the Final Velocity
v = a*t

Step 5: Calculate the Speed
Speed = |v| (magnitude of velocity)

Now, let's calculate the values step-by-step.

To find the speed of the block 3.50 seconds after it starts moving, we need to use Newton's second law of motion. This law states that the acceleration of an object is equal to the net force acting on it divided by its mass.

First, let's find the horizontal and vertical components of the force applied to the block by the cord.

The horizontal component of the force can be found using the equation: F_horizontal = F * cos(θ)
F_horizontal = 14.1 N * cos(19.0°) = 13.409 N (rounded to three decimal places)

The vertical component of the force can be found using the equation: F_vertical = F * sin(θ)
F_vertical = 14.1 N * sin(19.0°) = 4.592 N (rounded to three decimal places)

Next, we can calculate the acceleration of the block using Newton's second law. The net force acting on the block can be determined by using the horizontal component of the force, as there is no friction on the horizontal floor.

Since there is no friction, the net force is equal to the applied force (F_horizontal).

Using the equation: Net force = mass * acceleration
13.409 N = 6.00 kg * acceleration

Solving for acceleration:
acceleration = 13.409 N / 6.00 kg = 2.235 m/s^2 (rounded to three decimal places)

Now that we have the acceleration, we can find the speed of the block after 3.50 seconds by using the kinematic equation:

v = u + at

Where:
v = final velocity (speed)
u = initial velocity (assumed to be 0, as the block starts from rest)
a = acceleration
t = time

Plugging in the given values:
v = 0 + (2.235 m/s^2 * 3.50 s)
v = 7.823 m/s (rounded to three decimal places)

Therefore, the speed of the block 3.50 seconds after it starts moving is approximately 7.823 m/s.