Problem 50 - Precalculus : Building Concepts and Connections

Question

Round your answers to two decimal places. A golf ball is hit from a tee with a launch angle of \(13.2^{\circ}\) and speed 140 miles per hour. Express the velocity of the ball in component form. (Source: www.golf.com)

Step-by-Step Solution

Answer

Compute the expressions from step 3 to obtain the horizontal and vertical components of the velocity. This will represent the velocity of the golf ball in component form. The final values need to be rounded to two decimal places.

Step 1: Convert Speed to Meters Per Second

Given the speed in miles per hour (mph), we need to convert it into a unit which is standard in physics, that is, meters per second (m/s). This can be done by multiplication with a conversion factor: \(1 \text{ mph} = 0.44704 \text{ m/s}\). Hence, 140 mph would be equivalent to \(140 * 0.44704 \text{ m/s}\).

Step 2: Convert Launch Angle to Radians

Since the trigonometric functions in many calculators use units of radians, we need to convert the launch angle from degrees to radians using the conversion factor: \(1^{\circ} = \frac{\pi}{180} \text{ rad}\). Hence, \(13.2^{\circ}\) would be equivalent to \(13.2 * \frac{\pi}{180} \text{ rad}\).

Step 3: Compute the Horizontal and Vertical Components of the Velocity

Now we find the x-component (horizontal) and the y-component (vertical) of the velocity using the following formulas and using the results from step 1 and 2. \(V_x = V * \cos(\theta)\) (for the x-component), \(V_y = V * \sin(\theta)\) (for the y-component). Where V is the magnitude of velocity, and \(\theta\) is the launch angle.

Key Concepts

Converting Speed to Meters Per Second

Understanding how to convert speeds from one unit to another is essential when solving physics problems. In our example, we're converting a speed from miles per hour (mph) to meters per second (m/s), which is the standard unit of velocity in the System International (SI).

For conversion, we use the factor: \(1 \text{ mph} = 0.44704 \text{ m/s}\). The conversion is a simple multiplication problem. For a golf ball traveling at 140 mph, you would calculate the speed in m/s as: \(140 \times 0.44704 \text{ m/s}\).

This step is crucial because all subsequent calculations require the speed to be in m/s for consistency with the standard units of physics equations. Converting units may seem straightforward, but it is a common source of error if overlooked or done incorrectly. Always double-check your conversion factors to ensure accuracy in your results.

Launch Angle to Radians Conversion

Angles can be measured in degrees or radians. Most physics problems, including projectile motion, use radians because it simplifies calculations, especially when dealing with trigonometric functions.

The conversion from degrees to radians is done using the relationship: \(1^{\text{°}} = \frac{\text{π}}{180} \text{ rad}\). So, to convert our golf ball's launch angle of 13.2 degrees to radians, we multiply by this factor: \(13.2^{\text{°}} \times \frac{\text{π}}{180}\).

It's essential to perform this conversion before computing velocity components since trigonometric functions in calculators often require angle measures in radians for correct results.

Horizontal and Vertical Velocity Components

Breaking down velocity into horizontal (x-component) and vertical (y-component) parts is pivotal in analyzing projectile motions, such as a golf ball's flight. The process uses trigonometry, specifically the sine and cosine functions.

To find the x-component of the velocity (the horizontal speed), use the formula: \(V_x = V \times \text{cos}(\theta)\). Conversely, to find the y-component (the vertical speed), the formula is: \(V_y = V \times \text{sin}(\theta)\), where \(V\) is the speed of the golf ball in m/s and \(\theta\) is the launch angle in radians.

Using cosine gives us the proportion of velocity in the direction of the launch that is horizontal, while sine gives us the vertical proportion. These components are critical in determining how far and how high the ball will travel. Always make sure that your calculator is set to the correct angle mode (radians in most physics problems) to avoid incorrect values for these components.