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

문제 42b

2장, 문제 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?

검증된 단계별 안내

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.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

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주요 개념

질문에 올바르게 답하기 위해 반드시 이해해야 하는 핵심 개념들은 다음과 같습니다.

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.

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?

비슷한 문제에 대한 검증된 영상 답변:

주요 개념

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?