Problem 26 - University Physics with Modern Physics

Question

An athlete throws a ball of mass \(m\) directly upward, and it feels no appreciable air resistance. Draw a free-body diagram of this ball while it is free of the athlete's hand and (a) moving upward; \((b)\) at its highest point; (c) moving downward. (d) Repeat parts \((a),(b),\) and \((c)\) if the athlete throws the ball at a \(60^{\circ}\) angle above the horizontal instead of directly upward.

Step-by-Step Solution

Answer

The free-body diagrams for both throws have a single downward arrow for gravity in each case.

Step 1: Understanding Forces on the Ball Moving Upward

When the ball is moving upward, the only force acting on it is the gravitational force pulling it downward. In the free-body diagram, draw a single arrow pointing downward labeled as 'Weight' or 'Gravity'.

Step 2: At the Ball's Highest Point

At the highest point, the ball momentarily stops before starting to fall back down. The only force acting on it remains the gravitational force. In the free-body diagram, it is depicted the same as in Step 1: a single downward arrow labeled 'Weight' or 'Gravity'.

Step 3: Analyzing Forces on the Ball Moving Downward

While the ball is descending, the gravitational force continues to be the sole force acting on it. In the free-body diagram, again only draw one downward arrow, labeled 'Weight' or 'Gravity'.

Step 4: Forces on the Ball Thrown Upward at a 60-degree Angle (Moving Upward)

When the ball is thrown at an angle, while moving upward, the gravitational force still acts downward. In the free-body diagram, illustrate a downward arrow labeled 'Weight' or 'Gravity'. No other forces act due to no appreciable air resistance.

Step 5: At the Ball's Highest Point Under Angled Throw

At the apex of its trajectory, the ball pauses with only gravity pulling it downward. Draw this in the free-body diagram as a single arrow pointing downward, marked 'Weight' or 'Gravity'.

Step 6: Forces on the Ball Moving Downward After Angled Throw

As the ball falls back down, gravity remains the only force. This is represented in the free-body diagram by a single downward arrow labeled 'Weight' or 'Gravity'.

Key Concepts

Gravitational force

Gravitational force is the natural force of attraction between two masses. On Earth, it's what gives weight to physical objects and causes them to fall towards the ground when dropped. When you throw a ball upwards, whether directly or at an angle, its motion is constantly influenced by gravity. The gravitational force acts continuously towards the center of the Earth, and for simple projectile problems, it is often the only force considered if air resistance is negligible.

Key points about gravitational force include:

• It is a constant force pulling the object downward, regardless of the object's position or velocity.
• The magnitude of this force is given by the equation: \( F_g = m imes g \), where \( m \) is the mass of the object and \( g \) is the acceleration due to gravity, approximately \( 9.8 \text{ m/s}^2 \) on Earth.

In the case of the thrown ball, the gravitational force is illustrated in the free-body diagram as a single arrow pointing downward, labeled 'Weight' or 'Gravity'. This remains true throughout its flight: moving upward, at the peak, and moving downward.

Projectile motion

Projectile motion involves objects that are launched into the air and influenced by gravity alone. When an athlete throws a ball upward, it undergoes projectile motion. While the ball can be thrown directly upward or at angles like \( 60^{\circ} \), the principles of projectile motion determine its path.

Key characteristics of projectile motion include:

• A constant downward acceleration due to gravity affecting vertical motion.
• Horizontal and vertical motions are independent of each other. Horizontal movement has constant velocity if air resistance is neglected.
• The trajectory is parabolic, reaching a maximum height before descending back. At the apex, vertical velocity is momentarily zero.

In our exercise, throwing the ball at a \( 60^{\circ} \) angle showcases classic projectile motion: a combination of vertical rise and horizontal distance covered until the ball starts descending again.

Newton's laws of motion

Newton's laws of motion describe the fundamental principles governing the behavior of moving objects, and they are central to understanding projectile motion.- **First Law (Law of Inertia):** An object in motion stays in motion with the same speed and direction unless acted upon by an unbalanced force. In the absence of air resistance, a thrown ball will maintain its horizontal velocity due to inertia.- **Second Law (Law of Acceleration):** The acceleration of an object depends on the net force acting on it and is inversely proportional to its mass: \( F = m imes a \). Here, gravity provides the necessary force to decelerate the ball as it ascends and accelerate it as it descends.- **Third Law (Action and Reaction):** For every action, there's an equal and opposite reaction. While not directly applicable to the ball's flight, this law is evident when the ball is thrown—the force exerted by the hand is equal and opposite to the force experienced by the hand.In this exercise, Newton's laws help explain why the ball behaves as it does: it accelerates downward under gravity and, in the absence of other forces like air resistance, follows a predictable path.