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Chapter 4: Problem 72

An object is moving in the \(x-y\) plane with the position as a function of time given by \(\vec{r}=x(t) \hat{i}+y(t) \hat{j}\). Point \(O\) is at \(x\) \(=0, y=0\). The object is definitely moving towards \(O\) when (1) \(v_{x}>0, v_{y}>0\) (2) \(v_{x}<0, v_{y}<0\) (3) \(x v_{x}+y v_{y}<0\) (4) \(x v_{x}+y v_{y}>0\)

Short Answer

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Option (3): \(x v_x + y v_y < 0\).

Step by step solution

01

Understand Object's Movement

To determine if the object is moving towards point O at \((0,0)\), we must consider its position vector \(\vec{r} = x(t) \hat{i} + y(t) \hat{j}\). The object moves toward the origin if the direction of its velocity vector points in the opposite direction of its position vector.
02

Identify the Velocity Vector

The velocity vector \(\vec{v}\) is the derivative of the position vector with respect to time: \(\vec{v} = \frac{d}{dt} (x(t) \hat{i} + y(t) \hat{j}) = v_x \hat{i} + v_y \hat{j}\), where \(v_x = \frac{dx}{dt}\) and \(v_y = \frac{dy}{dt}\).
03

Determine Dot Product

Calculate the dot product \(x v_x + y v_y\) to see if it's negative. If \(x v_x + y v_y < 0\), then the angle between the position vector \(\vec{r}\) and the velocity vector \(\vec{v}\) is obtuse, meaning \(\vec{v}\) is directed opposite to \(\vec{r}\), indicating movement towards point O.
04

Choose the Correct Option

Since an obtuse angle between \(\vec{r}\) and \(\vec{v}\) implies movement towards the origin, the correct condition for the object moving towards O is when \(x v_x + y v_y < 0\). Therefore, the correct option is (3).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Kinematics
Kinematics is the branch of physics that deals with the motion of objects without considering the causes of motion. It focuses on describing how objects move with respect to time. Analyzing the motion involves understanding different quantities like position, velocity, and acceleration. These quantities can often be described using vectors, which gives us both the magnitude and direction of motion.
  • **Position** defines the specific point in space where the object is located relative to a reference point, usually defined as the origin.

  • **Velocity** is the rate of change of position with respect to time, showing how fast and in what direction an object is moving.

  • **Acceleration** is the rate of change of velocity with time, indicating how an object's velocity changes.
To solve kinematic problems, understanding these vector quantities and their relationships is crucial. The study of kinematics provides the foundation for more complex types of motion analysis encountered in physics.
Position Vector
The position vector is a fundamental concept in kinematics, serving as a mathematical representation of an object's location in space. It is depicted as a vector \(\vec{r}\) originating from the origin (0,0) and pointing towards the object's position at any given time.
  • In a two-dimensional space, it鈥檚 typically expressed as \(\vec{r} = x(t) \hat{i} + y(t) \hat{j}\) where \(x(t)\) and \(y(t)\) are functions of time, representing the coordinates of the object in the x-y plane.

  • The position vector provides information not only about the distance of the object from the origin, but also its specific location in relation to the coordinate system.
The position vector is crucial in determining motion details such as displacement, which is the change in the position vector over time. Being proficient in understanding position vectors is vital for solving problems related to an object's movement in space.
Velocity Vector
The velocity vector provides essential details about the object's rate of change of position and direction of movement. Derived from the position vector, it offers insights into the object's speed and heading.
  • Mathematically, the velocity vector \(\vec{v}\) is the derivative of the position vector \(\vec{r}\) with respect to time: \(\vec{v} = \frac{d}{dt}(x(t) \hat{i} + y(t) \hat{j}) = v_x \hat{i} + v_y \hat{j}\).

  • Here, \(v_x\) and \(v_y\) are the components of the velocity vector along the x and y axes, respectively, calculated as \(v_x = \frac{dx}{dt}\) and \(v_y = \frac{dy}{dt}\).
The velocity vector enables us to understand how quickly an object moves from one point to another and in which direction. Recognizing how to compute and interpret this vector is critical for predicting future motion and comprehending the dynamics of moving bodies.
Dot Product
The dot product is a powerful tool in vector algebra, especially useful in determining the relationship between two vectors. When analyzing an object's motion, dot products help in assessing whether the velocity vector approaches or departs from a position vector.
  • The dot product between two vectors \(\vec{A}\) and \(\vec{B}\) is given by \(\vec{A} \cdot \vec{B} = A_x B_x + A_y B_y + A_z B_z\).

  • A crucial aspect is that the dot product reflects the cosine of the angle between two vectors. Therefore, if \(x v_x + y v_y < 0\), it indicates that the angle between the position vector \(\vec{r}\) and velocity vector \(\vec{v}\) is greater than 90 degrees (obtuse), showing the motion towards the origin.
Using the dot product, one can easily determine the directional tendency of vectors within various kinematic scenarios. Mastering its application aids in comprehensively analyzing and predicting the behavior of objects in motion.

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