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Chapter 7: Problem 176

A particles is moving in \(x-y\) plane. At certain instant.of time, the components of its velocity and acceleration are as follows: \(v_{x}=3 \mathrm{~ms}^{-1}, v_{y}=4 \mathrm{~ms}^{-1}, a_{x}=2 \mathrm{~ms}^{-2}\), and \(a_{y}=1 \mathrm{~ms}^{-2}\). The rateof change of speed at this moment is a. \(\sqrt{10} \mathrm{~ms}^{-2}\) b. \(4 \mathrm{~ms}^{-2}\) c. \(\sqrt{5} \mathrm{~ms}^{-2}\) d. \(2 \mathrm{~ms}^{-2}\)

Short Answer

Expert verified
The rate of change of speed is \(2 \, \text{m/s}^2\) (Option d).

Step by step solution

01

Understand the Given Variables

We are given the components of velocity: \(v_x = 3 \, \text{m/s}\) and \(v_y = 4 \, \text{m/s}\), and components of acceleration: \(a_x = 2 \, \text{m/s}^2\) and \(a_y = 1 \, \text{m/s}^2\). We need to find the rate of change of speed, which is the scalar acceleration component.
02

Calculate the Speed

The speed, \(v\), of the particle is the magnitude of its velocity vector. It is calculated using the Pythagorean theorem: \[ v = \sqrt{v_x^2 + v_y^2} = \sqrt{3^2 + 4^2} = \sqrt{9 + 16} = \sqrt{25} = 5 \, \text{m/s} \]
03

Determine the Rate of Change of Speed

The rate of change of speed is the component of acceleration in the direction of the velocity, given by: \[ \frac{dv}{dt} = \frac{v_x \cdot a_x + v_y \cdot a_y}{v} \] Substituting the values, \(v_x = 3\), \(v_y = 4\), \(a_x = 2\), \(a_y = 1\), and \(v = 5\), we have: \[ \frac{dv}{dt} = \frac{3 \times 2 + 4 \times 1}{5} = \frac{6 + 4}{5} = \frac{10}{5} = 2 \, \text{m/s}^2 \]

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Key Concepts

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

Velocity Components
Velocity components are a way to break down the overall movement of a particle into more manageable parts, particularly in two-dimensional motion. By dissecting the velocity vector into horizontal and vertical components, represented as \(v_x\) and \(v_y\), calculations become simpler and clearer. In our problem, the components are given as \(v_x = 3 \, \text{m/s}\) and \(v_y = 4 \, \text{m/s}\).
  • Horizontal Component \( v_x \): This is the velocity in the x-direction, which is typically along the horizontal plane.
  • Vertical Component \( v_y \): This represents the velocity in the y-direction, usually the vertical plane.
These components can be combined using the Pythagorean theorem to find the overall velocity's magnitude. By decomposing the motion into components, it becomes easier to analyze complex movements and understand how each directional velocity affects the particle's motion.
Acceleration Components
Just like velocity, acceleration can also be broken down into components in the x and y axes, labeled as \(a_x\) and \(a_y\). These components provide insight into how the object's speed is changing in each direction. For our exercise, \(a_x = 2 \, \text{m/s}^2\) and \(a_y = 1 \, \text{m/s}^2\).
  • Horizontal Acceleration \( a_x \): This component measures how quickly the velocity in the horizontal direction is changing.
  • Vertical Acceleration \( a_y \): This denotes how the velocity is changing in the vertical direction.
Understanding these components is crucial for calculating the total change in velocity, as they help in finding the specific effect of acceleration in each direction on the particle's motion.
Magnitude of Velocity
The magnitude of velocity, often simply referred to as speed, is a scalar quantity representing how fast an object is moving, regardless of direction. It is determined by taking the root of the sum of the squares of its velocity components. Given our particle's velocity components, it's calculated as follows: \[ v = \sqrt{v_x^2 + v_y^2} = \sqrt{3^2 + 4^2} = 5 \, \text{m/s} \]
  • Why is Magnitude Important? The magnitude of velocity is crucial because it tells us how fast the particle is moving overall, without regard to the direction.
  • Relation to Components: While the components help us understand the motion in specific directions, the magnitude provides a holistic view of the particle's total speed.
Calculating the magnitude is essential, especially when determining the particle's rate of change in speed, as it helps in evaluating the overall acceleration effect on the particle's motion.

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Most popular questions from this chapter

A particle of small mass \(m\) is joined to a very heavy body by a light string passing over a light pulley. Both bodies are free to move. The total downward force on the pulley is a. \(>>m g\) b. \(4 \mathrm{mg}\) c. \(2 \mathrm{mg}\) d. \(\overline{m g}\)

Two skaters weighing in the ratio \(4: 5\) and \(9 \mathrm{~m}\) apart are skating on a smooth frictionless surface. They pull on a rope stretched between them. The ratio of the distance covered by them when they meet each other will be a. \(5: 4\) b. \(4: 5\) c. \(25: 16\) d. \(16: 25\)

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