Problem 62 - Fundamentals of Physics

Question

A golfer takes three putts to get the ball into the hole. The first putt displaces the ball \(3.66 \mathrm{~m}\) north, the second \(1.83 \mathrm{~m}\) southeast, and the third \(0.91 \mathrm{~m}\) southwest. What are (a) the magnitude and (b) the direction of the displacement needed to get the ball into the hole on the first putt?

Step-by-Step Solution

Answer

The required displacement is approximately 3.74 m at an angle of 3.82° west of north.

Step 1: Understand the Problem

The golfer makes three putts with given magnitudes and directions. We need to find the resultant displacement: the single vector that would accomplish the same total movement made by all three putts.

Step 2: Represent the Vectors

Translate the directional language into vector components:1. North (first putt): This vector is vertical with magnitude 3.66 m.2. Southeast (second putt): Direction gets broken down into components, forming a 45° angle from both the south and east axes; hence, using components of magnitude 1.83 m: - Component along South: \[1.83 \times \cos(45°) = \frac{1.83}{\sqrt{2}}\] - Component along East: \[1.83 \times \sin(45°) = \frac{1.83}{\sqrt{2}}\]3. Southwest (third putt): Similar breakdown with 45° angle; magnitude 0.91 m: - Component along South: \[0.91 \times \cos(45°) = \frac{0.91}{\sqrt{2}}\] - Component along West: \[0.91 \times \sin(45°) = \frac{0.91}{\sqrt{2}}\]

Step 3: Calculate the Resultant Components

Sum up the individual components for each direction:1. For the North-South axis: - Total north: 3.66 m in the north - Total south: components from SE and SW \[ \left(\frac{1.83}{\sqrt{2}} + \frac{0.91}{\sqrt{2}}\right)\] - Resultant vertical component = 3.66 - (above calculation for the south direction)2. For the East-West axis: - East from SE = \(\frac{1.83}{\sqrt{2}}\) - West from SW = \(\frac{0.91}{\sqrt{2}}\) - Resultant horizontal component = East - West component.

Step 4: Calculate the Magnitude of the Resultant Displacement

With the resultant components from step 3, use the Pythagorean theorem:\[R = \sqrt{(x_{resultant})^2 + (y_{resultant})^2}\]Where \(x_{resultant}\) and \(y_{resultant}\) are the net components along horizontal and vertical directions respectively.

Step 5: Calculate the Direction of the Resultant Vector

Find the angle \(\theta\) of the resultant vector with respect to the north using the tangent function:\[\theta = \tan^{-1}\left(\frac{x_{resultant}}{y_{resultant}}\right)\]This angle will give the direction as it forms with the north; adjust the angle direction precisely taking into account its placement in coordinate terms.

Step 6: Conclusion

The magnitude of the single displacement required is found using the Pythagorean theorem as explained, and the direction informs us how to aim the initial shot to achieve the same result as the sequence of three putts.

Key Concepts

Vector Components

Understanding vector components is crucial when dealing with vector displacement. When a golfer hits the ball in certain directions, each of these movements can be represented as vectors.

A vector illustrates both magnitude and direction. For practical use, we often break down vectors into components along predefined axes—usually North-South and East-West directions. This breakdown simplifies calculations and helps in understanding the cumulative effect of the different directions of movement.

For instance, a vector pointing directly north can be fully described by its magnitude with no additional components needed. However, vectors like those towards southeast or southwest require us to split them into two perpendicular components. By doing this, we resolve, or split, them into two simpler vectors along our chosen axes which allows for straightforward addition later on. Each component is calculated using trigonometric functions based on the angle from the main axes.

Resultant Displacement

To find a single, efficient movement that sums up all the separate putts, we seek the resultant displacement vector. This resultant vector accounts for the total effect of all initial movements, portraying it as a single vector.

By looking at the vector components for each putt, we begin by combining all components in the same axis direction: all north or south components are summed together and all east or west components are similarly added.

The resulting vector displacement is essentially the ‘final destination’ vector coming from merging all these components. This vector tells us both how far and in which direction the ball could have moved in one go instead of multiple putts.

Pythagorean Theorem

The Pythagorean theorem offers a straightforward way to calculate the magnitude of the resultant displacement.

When we have the resultant north-south and east-west components, these form the two shorter sides of a right triangle. The hypotenuse is equivalent to our resultant displacement vector.
So, the magnitude of this vector can be calculated using the formula:

• \(R = \sqrt{(x_{resultant})^2 + (y_{resultant})^2}\)

This systematically provides the total distance from the starting point to the hole, that the resultant movement would cover if it were carried out in one single motion rather than multiple steps.

Tangent Function

Once we know the magnitude of the resultant vector, the tangent function helps us unearth its direction. The angle of this vector with respect to a reference axis, usually the north direction, is found using inverse trigonometry.

The formula:

• \(\theta = \tan^{-1}\left(\frac{x_{resultant}}{y_{resultant}}\right)\)

This serves to find the precise angle \(\theta\)from the north axis. This is essential to determine exactly how to aim the ball so that it travels in a straight path to the hole. It brings us one step closer to mastering the art of finding precise vector directions in real-world scenarios.