Question

Concept Check After a tennis match the two players dash to the net to shake hands. If they both run with a speed of \(3 \mathrm{~m} / \mathrm{s}\), are their velocities equal? Explain.

Short answer

No, their velocities are not equal because they have different directions.

Step-by-step solution

Understand Velocity

Velocity is a vector quantity, which means it has both magnitude and direction. To determine if two velocities are equal, both the magnitude and the direction must be the same.

Analyze Given Information

We are given that both players run at a speed of \(3 \mathrm{~m/s}\). This is the magnitude of their velocities. However, the direction of their motion is not specified. We need to consider how direction affects velocity.

Consider Direction

As the players run towards each other to the net, their directions are opposite. One player runs in a positive direction (e.g., towards the net), while the other runs in a negative direction (e.g., also towards the net from the opposite side).

Compare Velocities

Since velocities are vector quantities, they have both magnitude and direction. Even though both players have the same speed (magnitude), their velocities are not equal because they are moving in opposite directions.

Key concepts

Vector Quantity

Velocity is a fascinating concept in physics, classified as a vector quantity. This implies that velocity has both size and direction, something crucial for understanding the exercise at hand.
While students often hear about speed, velocity adds a whole extra layer because direction plays a critical role. Unlike scalar quantities, which are defined by magnitude alone (like speed or distance), vector quantities like velocity provide complete information on how fast and where exactly an object is heading.
It's important to remember that when comparing velocities, both the magnitude and the direction must match for the velocities to be equal. This distinction is key to solving problems that involve moving bodies.

Magnitude

Magnitude refers to the "how much" aspect of quantity. In the case of velocity, magnitude is essentially the speed part. Think of it this way: when you say something is traveling at 3 meters per second, you are referring to the magnitude of the velocity.
Magnitude itself is a scalar, which means it only consists of a number and a unit. Compare this with our exercise, where both players have the same magnitude of velocity, running at 3 m/s each.
No matter the direction, if we only consider the magnitude, then we are merely expressing the speed. Remember, speed is the absolute value of velocity, and it does not tell us anything about the direction, just how fast something moves.

Direction

When discussing direction, we delve into the path that an object takes. Direction is what differentiates velocity from speed. Two objects may travel at the same speed or magnitude, yet if they face opposite directions, their velocities differ.
In the context of the exercise, the two tennis players run toward each other. Although their speeds are identical (3 m/s), their velocities differ due to their opposing directions.
Direction can significantly alter outcomes and meanings in physics problems. For instance, in a multi-dimensional space, specifying directions might include angles or reference points, adding layers to analysis.

Speed

Speed is a simple yet crucial concept frequently mentioned alongside velocity. It is the rate at which an object covers distance and is considered a scalar quantity.
Unlike velocity, speed doesn’t care about direction at all. It provides swift insight into how fast something is moving without any geographic or spatial considerations.
In our specific exercise, while the players run at a speed of 3 m/s, this doesn’t fully define their velocities. It’s essential to incorporate direction to make sense of velocity comparisons, which rely on the full vector quantity.