Question

As you walk briskly down the street, you toss a small ball into the air. (a) If you want the ball to land in your hand when it comes back down, should you toss the ball straight upward, in a forward direction, or in a backward direction? (b) Choose the best explanation from among the following: A. If you throw the ball straight up, you will leave it behind. B. You have to throw the ball in the direction in which you are walking. C. The ball moves in the forward direction at your walking speed at all times.

Short answer

Toss the ball straight upward. C is correct because the ball keeps your walking speed.

Step-by-step solution

Understand the Scenario

The exercise involves evaluating the motion of a ball relative to a person walking briskly. As the person walks, they toss the ball into the air and catch it on its return. The goal is to determine the correct direction to toss the ball so that it lands back in their hand.

Analyze the Relative Motion

When you throw the ball while walking, the ball retains a horizontal speed equal to your walking speed, due to inertia. Thus, whether you toss the ball straight up, forward, or backward, it will continue to move horizontally with you because it had the same horizontal velocity component as you at the moment of release.

Determine the Effect of Each Option

For option A, even if you throw the ball straight up, the horizontal velocity matches your speed, so it stays with you. For option B, throwing forward means adding extra horizontal speed, leading to the ball moving ahead. For option C, recognizing the ball's horizontal motion being unchanged by the throw accords with the laws of physics, allowing you to catch it.

Choose the Best Explanation

The correct answer is option C. The ball maintains the same horizontal velocity as you, which means it will land back into your hand if thrown straight up, assuming no other forces act in the horizontal direction. Thus, it is unnecessary to adjust the horizontal direction of the throw.

Key concepts

Relative Motion

Relative motion is an important concept in understanding how two objects move with respect to each other. When you walk briskly down the street and toss a ball into the air, both you and the ball are moving together horizontally. The ball's motion is relative to you, which means that from your point of view, the ball follows a predictable path.

When you initially throw the ball straight up, it has two components of motion: one vertical and one horizontal. The horizontal component is crucial here because it stays constant as long as no other forces interfere. This means that from your perspective, the ball travels straight up and then comes straight down, landing back in your hand.

This concept is crucial in physics because it shows how different velocities can interact. Understanding relative motion helps in various real-life applications, such as determining the correct direction and speed for catching a moving object.

Inertia

Inertia is a property of matter that causes it to resist changes in its state of motion. It is the reason why the ball retains its horizontal velocity when you throw it into the air.

Isaac Newton described inertia as an object's tendency to maintain its current motion unless acted upon by an external force. For example, when you toss the ball while walking, it already possesses the same horizontal velocity as your walking speed. Thanks to inertia, the ball continues to share this horizontal velocity during its entire journey through the air.

This concept explains why, even if you throw the ball straight up, it does not "fall behind," but instead returns to your hand. Inertia ensures that once the ball is moving, it continues to maintain that motion without deviation, unless something alters it, like a gust of wind or you changing speed.

Horizontal Velocity

Horizontal velocity is the speed and direction of an object's motion along the horizontal plane. When you toss the ball while moving, the ball inherits your walking speed as its horizontal velocity.

Despite being thrown vertically, the ball does not lose its horizontal motion. This is because there's no horizontal force acting on it to change this velocity, allowing it to keep moving alongside you.

It is essential to recognize that horizontal and vertical motions are independent of each other in projectile motion. This independence allows the ball to simultaneously move up and horizontally, ensuring it lands back into your moving hand. Calculating horizontal velocity involves understanding this isolated motion without external horizontal disruptions.

Projectile Motion

Projectile motion refers to the curved path that an object follows when it is thrown or propelled near the earth's surface and is subject only to gravity. When you throw the ball straight up while walking, it is subjected to projectile motion.

The path of the ball is a combination of its vertical and horizontal motions. Vertically, the ball moves up until gravity pulls it back down. Horizontally, it moves with the constant speed it had at the point of release due to inertia.

This behavior is typical of projectile motion: two simultaneous motions that do not influence each other. We solve such problems by considering these components separately. The ball's re-descent into your hand illustrates how projectile motion operates under these dual influences of gravity and initial velocity, providing an insightful way to predict motion that ignores air resistance.