Problem 30 - Fundamentals of Physics

Question

A soccer ball is kicked from the ground with an initial speed of \(19.5 \mathrm{~m} / \mathrm{s}\) at an upward angle of \(45^{\circ} .\) A player \(55 \mathrm{~m}\) away in the direction of the kick starts running to meet the ball at that instant. What must be his average speed if he is to meet the ball just before it hits the ground?

Step-by-Step Solution

Answer

Player must run at \(15.5\, \mathrm{m/s}\) to meet the ball.

Step 1: Analyze the Motion of the Ball

The ball is kicked at an initial speed of \(v_0 = 19.5\, \mathrm{m/s}\) at a \(45^\circ\) angle. The horizontal and vertical components of the initial velocity are equal due to this angle. Calculate these components using \(v_{0x} = v_0 \cos(45^\circ)\) and \(v_{0y} = v_0 \sin(45^\circ)\), which both equal \(\frac{19.5}{\sqrt{2}}\).

Step 2: Determine Time of Flight

The time for which the ball will be in the air (the time of flight) can be determined by using the formula \(t = \frac{2v_{0y}}{g}\), where \(g = 9.8\, \mathrm{m/s}^2\) is the acceleration due to gravity. Calculate \(t = \frac{2 \times \frac{19.5}{\sqrt{2}}}{9.8}\).

Step 3: Calculate Distance Traveled by the Ball Horizontally

The horizontal distance covered by the ball is given by \(d = v_{0x} \cdot t\). Substitute the time from Step 2 and the velocity component to find \(d\).

Step 4: Calculate the Time Required for the Player to Reach the Point

The player starts 55 meters away from the launch point of the ball. To meet the ball just as it lands, the player must reach this point in the same time it takes the ball to cover its horizontal distance. From Step 2, use this time \(t\) as the duration for which the player must run.

Step 5: Calculate the Required Speed of the Player

The average speed of the player is calculated as \(v = \frac{\text{distance}}{\text{time}}\). The player needs to cover 55 meters in the time determined in Step 2. Use this time to calculate the speed needed as \(v = \frac{55}{t}\).

Key Concepts

Initial Velocity

In projectile motion, the initial velocity is a critical component as it sets both the path and the coverage of the projectile. In this soccer ball scenario, the initial velocity is given as 19.5 meters per second at a launch angle of 45 degrees. This initial velocity can be broken down into two parts:

• Horizontal component ( \(v_{0x}\)): This dictates how far the ball travels across the field. It's calculated with \(v_{0x} = v_0 \cos(45^\circ)\). For our soccer ball, this means \(v_{0x} = \frac{19.5}{\sqrt{2}}\).
• Vertical component ( \(v_{0y}\)): This impacts how far and high the ball goes into the air, calculated using \(v_{0y} = v_0 \sin(45^\circ)\). Similarly, \(v_{0y} = \frac{19.5}{\sqrt{2}}\).

Both components are equal because of the 45-degree angle. This initial division helps us predict the path of the ball, aiding in further calculations for time and distance.

Time of Flight

The time of flight is the total period that the projectile, like our soccer ball, spends in the air. To determine this, we consider only the vertical motion and the effect of gravity. The formula used is:\[ t = \frac{2v_{0y}}{g} \]where \(g\) is the gravitational acceleration, 9.8 m/s². As we previously determined, the initial vertical velocity, \(v_{0y}\), is \(\frac{19.5}{\sqrt{2}}\).Calculating further:\[ t = \frac{2 \times \frac{19.5}{\sqrt{2}}}{9.8} \]This gives us a precise measure of how long the ball remains in the air, crucial for synchronizing the player's sprint to meet it just on the descending pathway.

Horizontal Distance

Horizontal distance in projectile motion refers to the total stretch covered across the ground. It's determined by the horizontal velocity and the time the projectile spend in the air. From our setup, the formula becomes:\[ d = v_{0x} \cdot t \]With the horizontal component \(v_{0x}\) as \(\frac{19.5}{\sqrt{2}}\), we place this into the equation using the time of flight calculated previously:Combine to find the actual distance the ball travels before touching back down. This calculation ensures we know the travel path across the field providing us with the necessary point for the player to intercept.

Average Speed

Determining the average speed necessary for the player to meet the projectile involves covering the initial gap between them in sync with the ball's trajectory. Given that the player starts 55 meters away and must reach the meeting point within the time of flight:The formula for average speed is:\[ v = \frac{\text{distance}}{\text{time}} \]Here, the 'distance' is 55 meters, and 'time' is identical to the ball's time of flight calculated via vertical motion. By solving:\[ v = \frac{55}{t} \]you grasp how quickly the player must move from their stationary start to align perfectly with the ball's landing spot, ensuring the necessary coordination essential in sports situational plays.