Problem 49 - Fundamentals of Physics

Question

A football kicker can give the ball an initial speed of \(25 \mathrm{~m} / \mathrm{s}\). What are the (a) least and (b) greatest elevation angles at which he can kick the ball to score a field goal from a point \(50 \mathrm{~m}\) in front of goalposts whose horizontal bar is \(3.44 \mathrm{~m}\) above the ground?

Step-by-Step Solution

Answer

To determine the angles, use the range formula and solve for angles that meet the height requirement using quadratic equations and trigonometry.

Step 1: Define the Problem

We need to determine the angles at which a kicker can kick a ball with an initial speed of \(25\, \text{m/s}\) from a distance of \(50\, \text{m}\) so that it just clears the goalposts that are \(3.44\, \text{m}\) above the ground.

Step 2: Use Projectile Motion Equations

Use the range equation for projectile motion: \( R = \frac{v_0^2 \sin(2\theta)}{g} \), where \(R = 50 \text{ m}\), \(v_0 = 25 \text{ m/s}\), and \(g = 9.8 \text{ m/s}^2\). Set \(R = 50\).

Key Concepts

Initial Velocity

When we talk about initial velocity in projectile motion, we're referring to the speed at which an object is launched into the air. In this exercise, the kicker imparts an initial speed of \( 25 \, \text{m/s} \) to the football. This initial velocity is crucial because it determines how far and how high the ball will travel. Without a sufficiently high initial velocity, the football won't reach the goalposts.

Initial velocity is often broken down into horizontal and vertical components using some trigonometry. These components help us understand different aspects of the projectile's motion:

• **Horizontal Component**: Determines how far the projectile travels. Given by \( v_{0x} = v_0 \cos(\theta) \).
• **Vertical Component**: Dictates the height the projectile reaches. Calculated as \( v_{0y} = v_0 \sin(\theta) \).

The angle \( \theta \) at which the ball is kicked affects both these components, which is why understanding the initial velocity is key to solving this problem.

Elevation Angle

The elevation angle is the angle above the horizontal at which the ball is kicked. In projectile motion, this angle significantly affects how far the object will travel and what trajectory it will follow. To get the ball over the goalpost, the kicker must choose appropriate elevation angles.

For any given initial velocity, there are usually two possible angles that achieve the same horizontal distance—symmetrical about \(45^\circ\). However, not both angles will clear obstacles of different heights. Therefore, in this scenario, understanding the concept of elevation angle is vital so that the football clears the 3.44 m high goalpost at 50 m away.

• **Least Elevation Angle**: The smallest angle which allows the ball to clear the obstacle.
• **Greatest Elevation Angle**: The largest angle that achieves the same horizontal distance while still clearing the obstacle.

Calculating the exact elevation angles involves using the projectile motion equations, considering factors such as initial speed and the height of goalposts.

Range Equation

The Range Equation is a crucial formula in projectile motion, used to calculate the horizontal distance a projectile will travel before landing. It is given by:\[R = \frac{v_0^2 \sin(2\theta)}{g}\]Here, \( R \) is the range or horizontal distance covered, \( v_0 \) is the initial velocity, \( \theta \) is the elevation angle, and \( g \) is the acceleration due to gravity—approximately \( 9.8 \, \text{m/s}^2 \).

The formula shows how both the initial speed and the angle of projection play integral roles in reaching the necessary distance. By determining the right combination of these variables using the range equation, the kicker can ensure the football travels precisely 50 meters, thus successfully reaching and clearing the goalposts.

• The **sin(2\(\theta\))** term in the equation reflects the impact of launch angle
• When \(\theta = 45^\circ\), the projectile achieves maximum range

Applying this equation helps the kicker understand the mathematical relationship between velocity, angle, and distance, allowing for the precise adjustments needed to score a field goal.