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Chapter 7: Problem 1

A soccer player is making a free kick and the opposing team is making a wall to protect their goal. Is it always theoretically possible for the kicker to hit the goal in an ideal situation with only a vertical acceleration due to gravity?

Short Answer

Expert verified
Yes, it is theoretically possible under ideal conditions with only vertical gravity.

Step by step solution

01

Understand the Problem

Read the problem carefully. The question asks whether it is always theoretically possible to hit the goal from a free kick, considering only vertical acceleration due to gravity.
02

Identify Known Variables

Recognize that gravity acts vertically on the soccer ball. The only force acting on the ball after being kicked is gravity, which affects the vertical component of the ball's velocity.
03

Horizontal Motion Consideration

Understand that the horizontal motion is not affected by gravity and remains constant. This means the ball travels horizontally in a straight line at a constant speed.
04

Vertical Motion Analysis

Analyze the vertical motion using the equation of motion under gravity: \( y(t) = v_{0y} t - \frac{1}{2} g t^2 \) where \( y(t) \) is the height, \( v_{0y} \) is the initial vertical velocity, \( g \) is the acceleration due to gravity, and \( t \) time.
05

Ideal Trajectory

In an ideal situation, the trajectory of the ball is parabolic, influenced only by its initial velocity and gravity. The kicker can adjust the initial velocity and angle to ensure the ball follows a desired path.
06

Theoretical Conclusion

Given ideal conditions (absence of air resistance and perfect control over the initial parameters), it is theoretically possible for the kicker to hit the goal. They can always find an appropriate angle and speed to ensure the ball's parabolic trajectory intersects the goal.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Vertical Acceleration Due to Gravity
Gravity is a crucial force in projectile motion. When a soccer player kicks a ball, gravity influences its vertical motion by pulling it towards the ground.
This force acts constantly and downwards, giving it a value of approximately 9.8 m/s² on Earth.

In the context of our exercise, the soccer ball experiences vertical acceleration due to gravity while it travels through the air. The vertical motion can be studied using equations by considering the downward pull of gravity.
Understanding this helps us predict the ball’s vertical position at any point during its flight.
Horizontal Motion in Physics
In projectile motion, horizontal and vertical movements are treated independently because gravity only affects the vertical component.
When a soccer ball is kicked, its horizontal motion stays constant if we ignore air resistance.

This means the ball moves horizontally with a constant speed, determined solely by the initial kick.
It's essential to remember that no force acts on the ball in the horizontal direction (assuming no friction or air drag), so its horizontal velocity remains unchanged throughout the flight.
This concept is key to understanding how far the ball will travel horizontally.
Ideal Projectile Motion
Ideal projectile motion refers to a situation where only gravity influences the motion of a projectile, like our soccer ball, with no other forces acting on it.
In this case, the ball follows a parabolic path determined by its initial speed and angle.

For a soccer free kick, the player needs to adjust these initial conditions to ensure the ball's trajectory reaches the goal.

The ideal trajectory helps us understand how different initial parameters (speed and angle) affect the ball's path, enabling the player to predict and control the ball's flight accurately.
This theoretical approach provides essential insights into hitting the target despite various real-world complications.
Equations of Motion Under Gravity
To predict the soccer ball's path, we use equations of motion that account for gravity's effect.
The main equation for vertical motion is: ewline y(t) = v_{0y} t - \frac{1}{2} g t^2 ewline Here, \(y(t)\) represents the height at time \(t\), \(v_{0y}\) is the initial vertical velocity, and \(g\) is the gravitational acceleration (9.8 m/s²).

This equation helps us calculate the ball's vertical position over time.
By understanding these equations, we can model and predict the ball's movement precisely.
This predictive power is fundamental for athletes and engineers alike.

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Most popular questions from this chapter

You tie a long, strong rope between your car and a tree in order to exert a large force on the car. How can you pull the rope to ensure that you pull at the car with a much larger force than you can exert on the rope?

In this project we will develop a model to determine the motion of a weather balloon released from the ground. We start from a simplified model and gradually add features to make to model more realistic. After the balloon is released, it is driven by buoyancy. Initially, we will assume that the buoyancy force is a constant, \(B\). (a) Draw a free-body diagram of the balloon. Identify the forces and introduce symbols. Indicate the relative magnitudes of the forces by the length of the vectors. First, let us neglect air resistance. (b) What is the acceleration of the balloon? (c) Find the position and velocity of the balloon as a function of time. Let us now introduce air resistance, using a quadratic law: \(\mathbf{F}_{D}=-D v \mathbf{v}\). (d) Show that the acceleration of the balloon in the upward \((z)\) direction is \(a_{z}=\) \((B / m)-g-(D / m)\left|v_{z}\right| v_{z}\), where \(v_{z}\) is the velocity in the \(z\)-direction. (e) Sketch the acceleration and velocity as a function of time for the model including air resistance. (f) Find the asymptotic (terminal) velocity of the balloon. The balloon is released on a windy day, with a wind blowing with a velocity \(\mathbf{w}=w \mathbf{i}\) along the horizontal \(x\)-axis. (g) How does the wind modify the air resistance force \(\mathbf{F}_{D}\) on the balloon? (h) Draw a free-body diagram for the balloon in this case. Indicate the magnitude of the forces by the relative lengths of the vectors. (i) Find an expression for the acceleration a of the balloon. What are the initial conditions for the motion of the balloon? (j) Why do we call the motion in the \(z\) and the \(x\) direction "coupled" in this case? Can you determine the motion of the balloon analytically? (k) We can determine the motion of the balloon using numerical methods. Write a program to find the velocity \(\mathbf{v}(t)\) and position \(\mathbf{r}(t)\) as functions of time. (It is sufficient to only include the main integration step in your answer - that is the part that determines \(\mathbf{v}(t+\Delta t)\) and \(\mathbf{r}(t+\Delta t)\) given \(\mathbf{v}(t)\) and \(\mathbf{r}(t)\). Figure \(7.16\) shows the result of a simulation. (l) Describe the motion of the balloon. Illustrate by relevant sketches. \((\mathbf{m})\) Find the asymptotic (terminal) velocity of the balloon. In a real situation, the wind velocity is smaller near the ground and increases gradually to the full velocity \(w_{0}\) as the balloon moves upward. Typically, the velocity of the wind can be described by \(\mathbf{w}=w_{0}(1-\exp (-z / d)) \mathbf{i}\), where \(d=10 \mathrm{~m}\) is a length determining the transition. (n) Rewrite your program to include this effect. (o) What is the terminal velocity of the balloon now?

The force from the Sun on the Earth acts directly towards the Sun, yet the Earth does not fall into the Sun. Explain.
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