91Ó°ÊÓ

When dealing with motion in two dimensions, such as with the hot air balloon, we encounter the concept of resultant velocity. Resultant velocity is a single vector that represents the combination of two or more velocity vectors. In our case, these vectors are the vertical rise of the balloon and the horizontal wind.

• The vertical speed of the balloon is given as 1.5 m/s upward.
• The horizontal speed due to the wind is adjusted to 2.78 m/s.

The resultant velocity can be visualized as the diagonal of a right triangle formed by these two perpendicular velocity components. The method to calculate it is by finding the hypotenuse of this triangle, which requires using the Pythagorean theorem.

In the scenario where two velocity components are perpendicular as they are here, the Pythagorean theorem becomes a handy mathematical tool. It allows us to find the magnitude of the resultant velocity vector which is the hypotenuse of a right triangle. The formula is:\[ v_{\text{resultant}} = \sqrt{(v_{\text{vertical}})^2 + (v_{\text{horizontal}})^2}\]For the given speeds:

• Vertical speed: 1.5 m/s
• Horizontal speed: 2.78 m/s

Substitute these values into the formula:\[ v_{\text{resultant}} = \sqrt{(1.5)^2 + (2.78)^2} \approx 3.16 \, \text{m/s}\]This calculation shows that the balloon actually moves with a combined speed of approximately 3.16 m/s.

Knowing how fast something moves is crucial, but knowing where it's going is equally important. Calculating the direction of motion involves finding the angle between the resultant velocity and one of the axes (usually the horizontal). This is done using trigonometric functions like tangent.We use the formula:\[ \tan(\theta) = \frac{\text{vertical speed}}{\text{horizontal speed}}\]With our known speeds, this becomes:\[ \tan(\theta) = \frac{1.5}{2.78} \approx 0.539\]By recognizing this, we solve for \( \theta \) by using the inverse tangent function (arctan):\[ \theta = \arctan(0.539) \approx 28.07^\circ\]This angle is measured above the horizontal, indicating that the balloon moves 28.07° upwards in the direction the wind blows.

Velocity in two dimensions can be broken down into simpler parts known as vector components. In vector math, these are typically horizontal and vertical components which make calculations more manageable. In our case:

• The vertical component (rising motion): 1.5 m/s.
• The horizontal component (wind motion): 2.78 m/s after conversion.

Vector components allow you to analyze the motion's impact in each dimension separately. They simplify calculating net effects using vector addition. Breaking the motion into components makes it straightforward to apply trigonometric functions and keep track of resultant directions, especially in diverse physics problems where various forces or influences act in different directions. This concept is vital in ensuring clarity and precision in identifying exact movement.