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Ch 03: Motion in Two or Three Dimensions
Young & Freedman Calc - University Physics 14th Edition
Young & Freedman Calc14th EditionUniversity PhysicsISBN: 9780321973610당신이 사용하는 게 아니라요?교과서 변경
모든 교과서Young & Freedman Calc 14th EditionCh 03: Motion in Two or Three Dimensions문제 7b
3장, 문제 7b

The coordinates of a bird flying in the xy-plane are given by x(t) = αt and y(t) = 3.0 m − βt2, where α = 2.4 m/s and β = 1.2 m/s2. Calculate the velocity and acceleration vectors of the bird as functions of time.

검증된 단계별 안내
1
Start by understanding the given position functions: x(t) = αt and y(t) = 3.0 m − βt². These functions describe the bird's position in the xy-plane at any time t.
To find the velocity vector, differentiate the position functions with respect to time. The velocity vector v(t) is composed of the derivatives of x(t) and y(t).
Calculate the derivative of x(t) = αt with respect to time t. This gives the x-component of the velocity: v_x(t) = d(x)/dt = α.
Calculate the derivative of y(t) = 3.0 m − βt² with respect to time t. This gives the y-component of the velocity: v_y(t) = d(y)/dt = -2βt.
To find the acceleration vector, differentiate the velocity components with respect to time. The acceleration vector a(t) is composed of the derivatives of v_x(t) and v_y(t). Since v_x(t) = α is constant, its derivative is zero. For v_y(t) = -2βt, the derivative with respect to time is a_y(t) = -2β.

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.
영상 길이:
2m

주요 개념

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

Velocity Vector

The velocity vector represents the rate of change of position with respect to time. It is derived by differentiating the position functions x(t) and y(t) with respect to time. For the bird, the velocity vector is v(t) = (dx/dt, dy/dt), which involves calculating the derivatives of x(t) = αt and y(t) = 3.0 m − βt².
추천 영상:
가이드 코스
06:44
Adding 3 Vectors in Unit Vector Notation

Acceleration Vector

The acceleration vector indicates the rate of change of velocity with respect to time. It is obtained by differentiating the velocity vector components. For the bird, the acceleration vector is a(t) = (d²x/dt², d²y/dt²), which requires calculating the second derivatives of the position functions x(t) and y(t).
추천 영상:
가이드 코스
06:44
Adding 3 Vectors in Unit Vector Notation

Differentiation

Differentiation is a mathematical process used to find the rate at which a quantity changes. In physics, it is essential for determining velocity and acceleration from position functions. For this problem, differentiate x(t) = αt and y(t) = 3.0 m − βt² to find the velocity, and differentiate again to find the acceleration.
추천 영상:
가이드 코스
13:04
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