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), where tt is in ss. What is the first time after t=0 st=0\text{ s} when the particle reaches a turning point?

Ch 02: Kinematics in One Dimension

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796당신이 사용하는 게 아니라요?교과서 변경

모든 교과서

Knight Calc 5th Edition

Ch 02: Kinematics in One Dimension

문제 42a

2장, 문제 42a

A particle's velocity is given by the function $v_{x} = (2.0 m/s) \sin (π t)$ , where $t$ is in $s$ . What is the first time after $t equals 0 s$ when the particle reaches a turning point?

검증된 단계별 안내

The turning point of a particle occurs when its velocity becomes zero, as this indicates a change in direction. Start by setting the velocity function 𝓋ₓ = (2.0 m/s) sin(πt) equal to zero.

Solve the equation (2.0 m/s) sin(πt) = 0. Since the coefficient 2.0 m/s is non-zero, the equation simplifies to sin(πt) = 0.

The sine function equals zero at integer multiples of π, i.e., when πt = nπ, where n is an integer. Simplify this to t = n, where n is an integer.

Since the problem asks for the first time after t = 0 s, choose the smallest positive integer value for n, which is n = 1. Substituting n = 1 into t = n gives t = 1 s.

Thus, the first time after t = 0 s when the particle reaches a turning point is t = 1 s. This is because the velocity changes sign at this point, indicating a reversal in direction.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

도움이 되었나요?

주요 개념

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

Velocity and Turning Points

Velocity is the rate of change of position with respect to time, and a turning point occurs when the velocity of a particle changes direction. This happens when the velocity function equals zero, indicating that the particle has momentarily stopped before reversing its direction.

Trigonometric Functions

The velocity function given is a sinusoidal function, specifically a sine function. Understanding the properties of sine functions, such as their periodic nature and the values at which they equal zero, is crucial for determining the times at which the particle reaches turning points.

Solving Equations

To find the first time after t = 0 s when the particle reaches a turning point, one must solve the equation 𝓋ₓ = 0. This involves setting the sine function to zero and finding the corresponding values of t, which requires knowledge of the sine function's periodicity and its specific angles where it equals zero.