Question

Determine If the velocity of object 1 relative to object 2 is \(v_{12}\) and the velocity of object 2 relative to object 3 is \(v_{23}\), what is the velocity of object 1 relative to object 3 ?

Short answer

The velocity of object 1 relative to object 3 is the sum: \( v_{13} = v_{12} + v_{23} \).

Step-by-step solution

Understanding the Relationship

To solve this problem, we need to understand the concept of relative velocity. The velocity of an object relative to another object is the difference between their individual velocities with respect to a common frame of reference.

Identify the Formula

The formula to find the relative velocity between two objects given their individual relative velocities to a third object is:\[ v_{13} = v_{12} + v_{23} \]where \( v_{13} \) is the velocity of object 1 relative to object 3.

Substitute the Known Values

In this exercise, you are provided with the velocities: \(v_{12}\) which is the velocity of object 1 relative to object 2, and \(v_{23}\) which is the velocity of object 2 relative to object 3. Simply substitute these into the formula:\[ v_{13} = v_{12} + v_{23} \]

Conclusion

The velocity of object 1 relative to object 3 is the sum of the velocity of object 1 relative to object 2 and the velocity of object 2 relative to object 3.

Key concepts

Velocity Addition

Velocity addition is a fundamental concept when dealing with the motion of objects in physics. It is crucial in understanding how different velocities interact in various scenarios. When adding velocities, we need to consider the direction and magnitude of each. The formula for adding velocities is straightforward. To find the resultant velocity of one object relative to another, we sum up their vector velocities. This process requires considering both speed and direction, as velocities are vector quantities. For instance, if object 1 is moving with a velocity of \(v_{12}\) relative to object 2, and object 2 is moving with a velocity of \(v_{23}\) relative to object 3, then the velocity of object 1 relative to object 3 is the sum of these two velocities: \(v_{13} = v_{12} + v_{23}\). This equation highlights that velocities can be added if they are taken in a consistent frame of reference. Remember: when envisioning velocity addition, think of it as combining directions and speeds to get the total effect of motion between objects.

Frame of Reference

The frame of reference is an essential concept in physics, especially when discussing relative motion. It acts as the viewpoint or perspective from which an object's motion is measured. Imagine it as an invisible platform from where you observe the movement of other objects. In our exercise, each object has a velocity relative to another, and the choice of frame of reference determines how these velocities are observed. A consistent frame of reference is necessary when solving problems involving relative velocity, as it helps to maintain a clear and standardized perspective. Here are key points about frames of reference:

• The choice of frame of reference can simplify or complicate your problem-solving.
• It can be moving or stationary, depending on the observer's position and speed.
• In general, no frame of reference is superior, but some can make calculations more direct.

Choosing the right frame of reference is a strategic part of analyzing the dynamics in motion-related problems.

Object Motion

Object motion refers to how an object changes its position over time, which is described by its velocity. Velocity indicates both the speed of the object and the direction in which it is moving. Understanding object motion requires acknowledging that motion is relative. An object might be stationary relative to one frame but moving relative to another. This relativity is where the concept of relative velocity enters, helping us understand how velocities can change depending on different observers or frames of reference. Important factors affecting object motion include:

• Initial position and speed of the object.
• Forces acting on the object, which can alter its speed and direction.
• Time, as it influences how far an object will travel and at what speed.

Studying object motion allows us to predict future positions and understand the dynamics of moving objects, providing a foundation for more advanced topics like kinematics and dynamics.