A particle's velocity is given by the function vx=(2.0 m/s)sin⁡(πt)\mathcal{v}_x = (2.0 \, \text{m/s}) \sin(\pi t)vx=(2.0m/s)sin(πt), where ttt is in sss. What is the particle's acceleration at that time?

Ch 02: Kinematics in One Dimension

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 02: Kinematics in One Dimension

Problem 42b

Chapter 2, Problem 42b

A particle's velocity is given by the function $$\mathcal{v}$_x = (2.0 \, $\text{m/s}$) $\sin$($\pi$ t)$ , where $t$ is in $s$ . What is the particle's acceleration at that time?

Verified step by step guidance

Identify the relationship between velocity and acceleration. Acceleration is the time derivative of velocity, so we need to compute the derivative of the given velocity function 𝓋ₓ(t).

Write the velocity function: 𝓋ₓ(t) = (2.0 m/s) sin(πt). To find acceleration, calculate the derivative of this function with respect to time t: aₓ(t) = d(𝓋ₓ)/dt.

Apply the derivative rule for sine functions: d(sin(πt))/dt = π cos(πt). Using this, the derivative of 𝓋ₓ(t) becomes aₓ(t) = (2.0 m/s) * π * cos(πt).

Simplify the expression for acceleration: aₓ(t) = (2.0π m/s²) cos(πt). This is the general formula for the particle's acceleration as a function of time.

To find the acceleration at a specific time, substitute the given time value into the formula aₓ(t) = (2.0π m/s²) cos(πt) and evaluate the result.

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Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Velocity Function

The velocity function describes how the velocity of a particle changes over time. In this case, the velocity is given by 𝓋ₓ = (2.0 m/s) sin (πt), indicating that the particle's velocity oscillates sinusoidally with time, where the amplitude is 2.0 m/s and the frequency is determined by the π factor.

Acceleration

Acceleration is the rate of change of velocity with respect to time. It can be calculated by taking the derivative of the velocity function. For the given velocity function, the acceleration will also be a function of time, reflecting how quickly the particle's velocity is changing at any given moment.

Differentiation

Differentiation is a fundamental concept in calculus used to find the rate of change of a function. In this context, differentiating the velocity function with respect to time will yield the acceleration function. This process involves applying the chain rule and trigonometric differentiation to obtain the correct expression for acceleration.

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A particle's velocity is given by the function vx=(2.0m/s)sin(πt)\(\mathcal{v}\)_x = (2.0 \, \(\text{m/s}\)) \(\sin\)(\(\pi\) t)vx=(2.0m/s)sin(πt), where ttt is in sss. What is the particle's acceleration at that time?

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Key Concepts

Velocity Function

Acceleration

Differentiation

A particle's velocity is given by the function $\(\mathcal{v}\)_x = (2.0 \, \(\text{m/s}\)) \(\sin\)(\(\pi\) t)$ , where $t$ is in $s$ . What is the particle's acceleration at that time?