Question

You toss a small ball vertically up in the air. How are the velocity and acceleration vectors of the ball oriented with respect to one another during the ball’s flight up and down?

Short answer

Question: Describe the orientation of the velocity and acceleration vectors for a ball tossed vertically up in the air during its flight up and down. Answer: During the flight up, the velocity vector is oriented upwards, while the acceleration vector (due to gravity) is oriented downwards. At the highest point, the velocity vector is momentarily zero, and the acceleration vector is still oriented downwards. During the flight down, both the velocity and acceleration vectors are oriented in the same direction, which is downwards.

Step-by-step solution

Understand the given problem

We are given a vertical motion for a small ball. And, we need to analyze this motion and determine the direction of the vectors for velocity and acceleration during the ball's flight up and down.

Understand Vertical Motion and Gravity

In a vertical motion, gravity is the only force acting on the ball. Gravity always acts downward or towards the Earth's center. The gravitational acceleration (g) is constant near the Earth's surface, which is approximately 9.8 m/s² directed downwards.

Analyzing Orientation of Velocity Vector During Flight Up

While the ball is moving upwards, the velocity vector is in the upward direction, and its magnitude is decreasing as it moves against the force of gravity. Acceleration due to gravity (g) is acting in the downward direction. Hence, during the flight up, the velocity and acceleration vectors are oriented in opposite directions.

Analyzing Orientation of Velocity Vector at the Highest Point

At the highest point in the motion, the ball decelerates to a velocity of 0 m/s. The velocity vector is momentarily zero. However, the acceleration due to gravity is still acting on the ball in the downward direction. Therefore, at the highest point, the velocity vector is zero, and the acceleration vector is completely in the downward direction.

Analyzing Orientation of Velocity Vector During Flight Down

As the ball starts descending, gravity begins to pull it downwards, and the velocity vector points in the downward direction. The acceleration due to gravity is still acting in the same downward direction. Hence, during the flight down, both the velocity and acceleration vectors are oriented in the same direction, which is downwards. To summarize, during the flight up, the velocity and acceleration vectors are oriented opposite each other, whereas during the flight down, they are oriented in the same direction.

Key concepts

Vertical Motion

Understanding vertical motion is crucial in analyzing various real-world scenarios, such as the flight of a ball thrown upwards. Vertical motion refers to the movement of an object in the upward or downward direction along a straight line. This simplified analysis assumes the absence of air resistance, focusing solely on the influence of gravity.

When tossing a ball upward, it initially moves away from the Earth's surface. The path of this motion is straightforward—it ascends, decelerates to a pause at the peak, and then accelerates back down. At the start, the ball's velocity is at its maximum and it gradually decreases as it fights against Earth's gravitational pull, eventually reaching zero at the zenith of its journey. Once it begins to descend, it picks up speed due to the same gravitational force that slowed it down.

During its ascent, the ball's upward velocity is at odds with gravity's tug, which eventually overpowers the initial velocity, causing the ball to stop and change direction. Thus, the challenge in vertical motion is not just to predict how high or how long an object will move, but also to understand how gravity continuously acts to modify its vertical velocity.

Acceleration due to Gravity

Acceleration due to gravity is a concept that is often misunderstood, as it can seem counterintuitive that an object in freefall accelerates even when it appears to be simply dropping downward. However, on Earth, any object experiencing only the force of gravity (neglecting air resistance) is subject to a consistent acceleration, typically approximated as \( g = 9.8 \text{ m/s}^2 \).

This acceleration is directed towards the center of the Earth and affects all objects equally, regardless of their mass. This means whether you drop a feather or a bowling ball in a vacuum, they will accelerate at the same rate (though in practice, air resistance significantly alters a feather's behavior). It's essential to recognize that this gravitational acceleration occurs at all times during the object's flight, not just on the way down. It contributes to the velocity change from the initial upward motion to the eventual downward fall, showcasing nature's consistent and predictable influence on objects in vertical motion.

Velocity and Acceleration Vectors

Vectors are mathematical entities used to represent quantities with both magnitude and direction, making them perfect tools for describing motion. In the context of projectile motion, velocity and acceleration vectors provide a clear picture of how an object's movement changes over time.

Let's look at the thrown ball again. Initially, as the ball is thrown upward, its velocity vector—indicating the ball's speed and direction—points upwards. However, the acceleration vector, due to gravity, always points downward. These two vectors are opposite in direction during the ball's ascent, which signifies that the ball is slowing down.

At the very top of the ball's path, the velocity vector momentarily vanishes (\( \vec{v} = 0 \) m/s), as the ball switches from moving upward to falling downward. However, the acceleration vector due to gravity remains downward and constant in magnitude. When the ball begins its descent, both the velocity and acceleration vectors align in the same downward direction, reflecting that the ball is speeding up as it falls back to Earth. By analyzing these vectors, one can predict not just the ball's current state but also its future behavior during flight.