91Ó°ÊÓ

Understanding velocity change is crucial when analyzing motion. Velocity is a vector that has both magnitude and direction. In the exercise, we dealt with a drone moving on a frictionless surface. Initially, the drone had a velocity of \(3.00 \hat{\mathbf{i}}\, \text{m/s}\) and later it changes to \(9.00 \hat{\mathbf{i}} + 4.00 \hat{\mathbf{j}}\, \text{m/s}\) after 10 seconds. To find out how the velocity changed, we subtract the initial velocity vector from the final velocity vector. This operation gives us the change in velocity:

• Initial velocity \(\vec{v}_i = 3.00 \hat{\mathbf{i}}\, \text{m/s}\)
• Final velocity \(\vec{v}_f = 9.00 \hat{\mathbf{i}} + 4.00 \hat{\mathbf{j}}\, \text{m/s}\)

Thus, the change in velocity \(\Delta \vec{v} = (9.00 - 3.00) \hat{\mathbf{i}} + 4.00 \hat{\mathbf{j}}\, \text{m/s} = 6.00 \hat{\mathbf{i}} + 4.00 \hat{\mathbf{j}}\, \text{m/s}\). This indicates that the drone's speed increased in both the \(\hat{\mathbf{i}}\) and \(\hat{\mathbf{j}}\) directions.

Acceleration measures how quickly velocity changes over time. According to Newton's second law, this concept relates directly to the application of force. To determine acceleration, we divided the velocity change by the time interval.

• Change in velocity \(\Delta \vec{v} = 6.00 \hat{\mathbf{i}} + 4.00 \hat{\mathbf{j}}\, \text{m/s}\)
• Time interval \(\Delta t = 10.0\, \text{s}\)

The formula for acceleration is \(\vec{a} = \frac{\Delta \vec{v}}{\Delta t}\). After calculating, we find that \(\vec{a} = 0.60 \hat{\mathbf{i}} + 0.40 \hat{\mathbf{j}}\, \text{m/s}^2\). This value indicates the drone accelerates differently in each direction. The \(\hat{\mathbf{i}}\) direction experiences a greater acceleration than the \(\hat{\mathbf{j}}\) direction.

Force components are key to understanding how force acts on an object in different directions. Using Newton's second law \(\vec{F} = m \cdot \vec{a}\), we can determine these components. The drone, with a mass of 1.50 kg, experiences acceleration calculated previously.

• Mass \(m = 1.50\, \text{kg}\)
• Acceleration \(\vec{a} = 0.60 \hat{\mathbf{i}} + 0.40 \hat{\mathbf{j}}\, \text{m/s}^2\)

The force components are found by multiplying mass by each acceleration component:

• \(F_x = 1.50 \times 0.60 = 0.90\, \text{N}\)
• \(F_y = 1.50 \times 0.40 = 0.60\, \text{N}\)

These show how the force is distributed in the horizontal planes, affecting the drone's movement.

The magnitude of a force combines its components into a single value, showing the total effect of the force. It's determined using the Pythagorean theorem in the context of vector components. From earlier, we found:

• Force components: \(F_x = 0.90\, \text{N}\) and \(F_y = 0.60\, \text{N}\)

To find the magnitude, use the equation:\[ F = \sqrt{F_x^2 + F_y^2} \]Substituting the values:\[ F = \sqrt{(0.90)^2 + (0.60)^2} \]\[ F = \sqrt{0.81 + 0.36} \]\[ F = \sqrt{1.17} \]\[ F \approx 1.08\, \text{N} \]Thus, the total magnitude of the constant force acting on the drone is approximately 1.08 N. This metric helps understand the overall impact of forces on the drone's motion.