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Chapter 4: Problem 13

Suppose that the acecleration voctor of a partick moving in the \(x-y\) planc is $$ a=3 i+2 j $$ where the acceleration is measurcd in \(\mathrm{m} / \mathrm{s}^{2}\). The velocity vector and the povition vectur are rero at \(t=0\). (a) What is the velocity vector of this particle as a finction of time? (b) What is the position vector as a function of time?

Short Answer

Expert verified
(a) \( v(t) = 3t\mathbf{i} + 2t\mathbf{j} \); (b) \( p(t) = \frac{3}{2}t^2\mathbf{i} + t^2\mathbf{j} \).

Step by step solution

01

Understanding the Acceleration Vector

The problem provides an acceleration vector \( a = 3\mathbf{i} + 2\mathbf{j} \) measured in \(\mathrm{m/s^2}\). This means the acceleration in the x-direction is 3\(\mathrm{m/s^2}\) and in the y-direction is 2\(\mathrm{m/s^2}\). We'll use this information to determine the velocity and position vectors.
02

Finding the Velocity Vector (Integration of Acceleration)

Velocity is the integral of acceleration. Since the initial velocity is zero at \(t=0\), integrate the components of the acceleration vector separately: \( v_x = \int 3 \, dt = 3t + C_x \) and \( v_y = \int 2 \, dt = 2t + C_y \). With initial velocities zero, \( C_x = 0 \) and \( C_y = 0 \). Thus, the velocity vector is \( v(t) = 3t\mathbf{i} + 2t\mathbf{j} \).
03

Finding the Position Vector (Integration of Velocity)

Position is the integral of velocity. Using the velocity components: \( p_x = \int 3t \, dt = \frac{3}{2}t^2 + C_x \) and \( p_y = \int 2t \, dt = t^2 + C_y \). Since the initial position is zero, \( C_x = 0 \) and \( C_y = 0 \). Therefore, the position vector is \( p(t) = \frac{3}{2}t^2\mathbf{i} + t^2\mathbf{j} \).

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Key Concepts

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

Acceleration Vector
The acceleration vector tells us how quickly a particle's velocity changes in each direction per unit of time. In our exercise, the given acceleration vector is \( a = 3\mathbf{i} + 2\mathbf{j} \). This means:
  • The rate of change of velocity in the x-direction is \( 3 \, \mathrm{m/s}^2 \).
  • The rate of change of velocity in the y-direction is \( 2 \, \mathrm{m/s}^2 \).
Acceleration vectors are crucial in dynamic systems to predict velocity and position over time. By integrating the acceleration vector, we can determine the velocity vector at any given time.
Velocity Vector
The velocity vector describes how fast the particle moves in each direction. Initially, the velocity vector is zero since it's at rest at \( t = 0 \). To find the velocity as a function of time, integrate the acceleration vector components:
  • In the x-direction: \( v_x = \int 3 \, dt = 3t + C_x \). Since initial velocity is zero, \( C_x = 0 \).
  • In the y-direction: \( v_y = \int 2 \, dt = 2t + C_y \). Here too, \( C_y = 0 \), due to zero initial velocity.
Thus, the velocity vector becomes \( v(t) = 3t\mathbf{i} + 2t\mathbf{j} \). This tells us how the particle's position changes over time.
Position Vector
The position vector indicates the location of the particle in the plane at any time. For our problem, this vector is initially zero as the particle starts from the origin. We calculate the position by integrating the velocity vector:
  • In the x-direction: \( p_x = \int 3t \, dt = \frac{3}{2}t^2 + C_x \). With the initial position zero, \( C_x = 0 \).
  • In the y-direction: \( p_y = \int 2t \, dt = t^2 + C_y \). The initial position is zero, so \( C_y = 0 \).
Thus, the position vector is \( p(t) = \frac{3}{2}t^2\mathbf{i} + t^2\mathbf{j} \), showing the particle's path in the \( x-y \) plane.
Integration
Integration is a key mathematical operation used to find functions describing accumulated quantities, like velocity and position, from their rates of change, such as acceleration. In this exercise, we:
  • First, integrated the acceleration vector to determine the velocity vector.
  • Then, integrated the velocity vector to find the position vector.
Through integration, constants of integration \( C_x \) and \( C_y \) are determined by initial conditions: zero for both velocity and position at \( t = 0 \). This process connects the concepts of acceleration, velocity, and position comprehensively.
Kinematics
Kinematics is the study of motion without considering forces that cause it. This exercise delves into kinematics by using vectors and integration to analyze the movement of a particle.
  • The acceleration vector provides the initial description of motion.
  • By integrating this, we calculate the velocity vector, showing motion speed and direction.
  • Further integration reveals the position vector, outlining the particle's trajectory over time.
In essence, kinematics allows us to map out the motion of the particle with respect to time, leading to greater insight into its overall movement.

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