Question

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 given by \( v_{13} = v_{12} + v_{23} \).

Step-by-step solution

Understanding Relative Velocity

The concept of relative velocity helps us determine how fast one object is moving compared to another. When you have velocities of different objects given relative to each other, you can combine them to find the velocity of an object relative to a reference point.

Applying the Relative Velocity Formula

To find the velocity of object 1 relative to object 3, we need to add the velocity of object 1 relative to object 2 ( v_{12} ) to the velocity of object 2 relative to object 3 ( v_{23} ). This is expressed as: \[ v_{13} = v_{12} + v_{23} \]

Finalizing the Expression

There is nothing more to simplify or calculate since we are only dealing with expressions of velocity. Thus, the resulting expression for the velocity of object 1 relative to object 3 remains: \[ v_{13} = v_{12} + v_{23} \]

Key concepts

Velocity Addition

To understand how to compute the velocity of an object in terms of another object, we use the concept of velocity addition. This technique allows us to combine velocities across different objects in motion. The formula for velocity addition is crucial when dealing with multiple reference frames. Consider two objects, where the velocity of object 1 relative to object 2 is known as \(v_{12}\), and the velocity of object 2 relative to another object 3 is \(v_{23}\). To find the velocity of object 1 relative to object 3, you simply add these velocities: \[ v_{13} = v_{12} + v_{23} \] This addition works under the assumption that all vectors are in one-dimensional motion and align perfectly. It's essential to ensure that all velocities you're considering are in the same direction before adding them.

Reference Frames

When solving physics problems, understanding and selecting the correct reference frames can make a big difference. A reference frame is essentially a perspective from which you measure velocity and position. By choosing different reference frames, the observed velocity of objects can change, which influences how we analyze motion.

• A reference frame could be stationary, like the ground.
• It could also be another moving object, like a car traveling down a highway.

In the context of the problem, if you're standing on object 2, you see object 1 move at \(v_{12}\). But if you switch to standing on object 3, you need to account for object 2's motion relative to object 3 as well. This is why reference frames are crucial; they dictate how velocities are perceived and how we apply the velocity addition rule properly.

Physics Problem Solving

Solving physics problems involves a clear understanding of concepts and the careful application of principles. Let's break down how to effectively solve problems involving velocity.
When you encounter a relative velocity problem:

• First, identify all involved objects and their respective motions.
• Determine the known velocities and which objects they correspond to.
• Decide on your reference frame. It's sometimes helpful to sketch a diagram showing the directions involved.

Once you have a clear view, apply velocity addition rules to find the desired unknown velocity. This structured approach not only helps in solving individual problems but also builds strong foundational skills for more complex physics challenges. Remember that practice in identifying and working through physics problems is key to mastering this critical thinking process.