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

Answer: During the upward flight, the velocity vector points upward and the acceleration vector (due to gravity) points downward. At the highest point, the velocity vector is momentarily zero (no direction) while the acceleration vector points downward. During the downward flight, both the velocity and acceleration vectors point downward.

Step-by-step solution

Identify the given information

The exercise tells us that a ball is tossed vertically up in the air. The only acceleration acting on the ball is due to gravity. The acceleration due to gravity has a constant value, and it points downwards.

Analyze the upward flight of the ball

During the upward flight, the velocity vector of the ball is pointing upward since the ball is moving up. However, the acceleration due to gravity is still acting downward. So during this phase, the two vectors have opposite directions.

Analyze the highest point of the flight

At the highest point of the ball's flight, the velocity vector becomes momentarily zero since the ball is neither moving up nor down. The acceleration due to gravity remains constant and points downward. At this point, the velocity vector can be regarded as having no direction, while the acceleration vector has a downward orientation.

Analyze the downward flight of the ball

During the downward flight, the velocity vector of the ball is pointing downward since the ball is moving in that direction. At the same time, the acceleration due to gravity continues to act downward. In this phase, the two vectors have the same direction.

Summarize the orientation of velocity and acceleration vectors

To answer the exercise, we have analyzed the orientation of the velocity and acceleration vectors throughout the ball's flight: 1. During the upward flight, the velocity vector points upward while the acceleration due to gravity points downward. They have opposite directions. 2. At the highest point, the velocity vector is momentarily zero (no direction) while the acceleration due to gravity is still downward. 3. During the downward flight, both the velocity and acceleration vectors point downward. They have the same direction.

Key concepts

Velocity

In the context of kinematics, the term **velocity** refers to the speed of an object in a particular direction. It’s a vector quantity, which means it has both magnitude and direction.
Imagine tossing a ball vertically into the air:

• When the ball travels upward, its velocity is directed obstructively towards the sky.
• At the peak of its trajectory, the ball momentarily stops, leading to a momentary velocity of zero.
• On its way back down, the velocity reverses direction, now aligning towards the earth.

Understanding velocity is crucial in analyzing how an object moves, as it shows us not only how fast an object is moving but also in which direction. The scenarios change with time as the ball moves upwards, stops briefly, and then descends. Each of these provides chances to observe velocity in action.

Acceleration

Acceleration describes how quickly the velocity of an object is changing. It is also a vector and can change in terms of direction, speed, or both.

• For a ball tossed in the air, its acceleration is consistently affected by one main force: gravity.
• As the ball rises, it slows down due to the gravitational pull acting in the opposite direction.
• Upon reaching the highest point, acceleration doesn’t cease; it continues working downward, preparing the ball to descend.
• During the downward motion, acceleration and velocity work together, speeding the ball’s descent.

Understanding the influence of acceleration helps predict how quickly and in which way an object’s velocity will change.

Gravity

**Gravity** is a force that attracts two bodies towards one another, usually noticeable as the attraction of objects towards the earth. It plays a pivotal role in our daily experiences with motion.

• For a vertically thrown ball, gravity is the key player directing its path after leaving your hand.
• Regardless of the ball moving up or down, gravity’s pull is unwavering, consistently acting downward at approximately 9.81 m/s² near the earth's surface.
• This constant downward acceleration causes the ball to slow down as it moves upward and speed up as it descends.

Knowing how gravity works allows one to analyze motion more thoroughly, predicting how objects thrown into the air, like our ball, behave throughout their journey.