91Ó°ÊÓ

Kinematic equations are essential in analyzing the motion of objects. They describe the relationships between various motion parameters like velocity, acceleration, displacement, and time. In our context, we are particularly interested in the equation for vertical motion, as we want to find out how long the ball stays in the air. The most commonly used kinematic equation for vertical motion is:\[ y = v_{y0} t - \frac{1}{2} g t^2 \]In this equation:

• \(y\) is the vertical displacement from the starting point.
• \(v_{y0}\) is the initial vertical velocity component, calculated by multiplying the total initial velocity by the sine of the angle of projection.
• \(g\) represents the acceleration due to gravity, usually \(9.8 \, \text{m/s}^2 \).
• \(t\) is the time the ball is in the air.

By using this equation, we can substitute known values to find out how long a projectile, such as the golfer's ball, will stay airborne. This step involves basic algebraic manipulation which helps determine the time with a little calculation involving solving a quadratic equation.

Trigonometry is vital in physics, especially when dealing with angles, because it allows us to break down vectors into components. In the problem of the golfer, the ball's initial velocity must be separated into horizontal and vertical components to analyze its path accurately. Here's how we apply trigonometry:- **Horizontal Component**: - Calculated as \( v_{x0} = v_0 \cos(\theta) \) - Here, \( v_0 \) is the initial velocity (46.0 m/s) and \( \theta \) is the launch angle (35.0°). - Plug these into the equation: \( v_{x0} \approx 46.0 \times \cos(35.0^{\circ}) \approx 37.7 \, \text{m/s} \).- **Vertical Component**: - Calculated as \( v_{y0} = v_0 \sin(\theta) \) - Again using \( v_0 = 46.0 \, \text{m/s} \) and \( \theta = 35.0^{\circ} \). - The equation becomes: \( v_{y0} \approx 46.0 \times \sin(35.0^{\circ}) \approx 26.4 \, \text{m/s} \).Through this breakdown, we can now straightforwardly use these components to plug into kinematic equations. This method shows how trigonometry enables effective decomposition of forces or velocities, converting complex motion into simpler, more manageable calculations.

In physics, particularly when dealing with motion that involves quadratic equations, the quadratic formula is a powerful tool. It's especially useful when determining time intervals in projectile motion, like in our golfer's problem.The quadratic formula is expressed as:\[ t = \frac{-b \pm \sqrt{b^2 - 4ac}}{2a} \]Here's the breakdown:

• \(a\), \(b\), and \(c\) are coefficients from the quadratic equation formed by substituting into the kinematic equation.
• The discriminant \((b^2 - 4ac)\) determines the nature of roots. A positive discriminant implies real and different roots, which is what we need.
• The formula results in two possible values for time \(t\). In physics, it's often the positive root that's relevant since negative time doesn't make sense.

For the golfing problem, after substituting into our kinematic equation and simplifying, the quadratic elements are \(a = -4.9\), \(b = 26.4\), and \(c = -5.5\). Plug these into the quadratic formula gives the time the ball takes to reach the green. By solving this, we determined that the time was approximately 5.39 seconds, using only the viable, positive result.