A particle's trajectory is described by $x = (\frac{1}{2} t^{3} - 2 t^{2}) m$ where $t$ is in $s$ . What are the particle's position and speed at $t equals 0 s$ and $t equals 4 s$ ?

검증된 단계별 안내

Step 1: Understand the problem. The particle's trajectory is given by two equations: x = (1/2t^2 - 2t^2) m and y = (1/2t^2 - 2t) m. We need to find the particle's position (x, y) and speed at two specific times: t = 0 s and t = 4 s.

Step 2: Calculate the position at t = 0 s. Substitute t = 0 into the equations for x and y. For x, substitute t = 0 into x = (1/2t^2 - 2t^2). For y, substitute t = 0 into y = (1/2t^2 - 2t). This will give the coordinates (x, y) at t = 0 s.

Step 3: Calculate the position at t = 4 s. Substitute t = 4 into the equations for x and y. For x, substitute t = 4 into x = (1/2t^2 - 2t^2). For y, substitute t = 4 into y = (1/2t^2 - 2t). This will give the coordinates (x, y) at t = 4 s.

Step 4: Determine the speed at t = 0 s and t = 4 s. To find the speed, calculate the magnitude of the velocity vector. First, find the velocity components by differentiating x and y with respect to t. For x, differentiate x = (1/2t^2 - 2t^2) to get v_x. For y, differentiate y = (1/2t^2 - 2t) to get v_y. Then, calculate the speed using the formula: speed = sqrt(v_x^2 + v_y^2).

Step 5: Evaluate the speed at t = 0 s and t = 4 s. Substitute t = 0 and t = 4 into the expressions for v_x and v_y obtained in Step 4. Use these values to calculate the speed at each time using the formula: speed = sqrt(v_x^2 + v_y^2).

비슷한 문제에 대한 검증된 영상 답변:

이 영상 해법은 위 문제에 도움이 된다고 튜터들이 추천한 것입니다.

영상 길이:

11m

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주요 개념

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

Kinematics

Kinematics is the branch of physics that describes the motion of objects without considering the forces that cause the motion. It involves concepts such as position, velocity, and acceleration, which are essential for analyzing the trajectory of a particle. In this question, understanding kinematics allows us to determine the particle's position and speed at specific time intervals.

Position and Displacement

Position refers to the location of a particle in space at a given time, while displacement is the change in position from one point in time to another. The equations provided describe the particle's position in two dimensions (x and y) as functions of time. Evaluating these equations at specific times (t = 0 s and t = 4 s) will yield the particle's exact position in the coordinate system.

Velocity

Velocity is a vector quantity that describes the rate of change of position with respect to time. It is calculated as the derivative of the position function with respect to time. In this problem, finding the velocity at the specified times involves differentiating the position equations to determine how fast and in what direction the particle is moving at those moments.