A particle's trajectory is described by x=(12t3−2t2) mandy=(12t2−2t) m,x = \left(\frac{1}{2} $$t^3$$ - $$2t^2$$\right) \, \text{m} \quad \text{and} \quad y = \left(\frac{1}{2} $$t^2 - 2t$$\right) \, \text{m},x=(21t3−2t2)mandy=(21t2−2t)m, where ttt is in sss. What is the particle's direction of motion, measured as an angle from the xx-axis, at t=0 st=0\text{ s}t=0 s and t=4 st=4\text{ s}t=4 s?

Ch 04: Kinematics in Two Dimensions

Knight Calc - Physics for Scientists and Engineers 5th Edition

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Knight Calc 5th Edition

Ch 04: Kinematics in Two Dimensions

Problem 9b

Chapter 4, Problem 9b

A particle's trajectory is described by $x = $\left$($\frac{1}{2}$ t^3 - 2t^2$\right$) \, $\text{m}$ $\quad$ $\text{and}$ $\quad$ y = $\left$($\frac{1}{2}$ t^2 - 2t$\right$) \, $\text{m}$,$ where $t$ is in $s$ . What is the particle's direction of motion, measured as an angle from the $x$ -axis, at $t=0$\text{ s}$$ and $t=4$\text{ s}$$ ?

Verified step by step guidance

Step 1: Understand the problem. The trajectory of the particle is given by the parametric equations x(t) = (1/2 t^2 - 2t^2) m and y(t) = (1/2 t^2 - 2t) m. The goal is to find the direction of motion, measured as an angle θ from the x-axis, at t = 0 s and t = 4 s. The direction of motion is determined by the velocity vector, which is derived from the time derivatives of x(t) and y(t).

Step 2: Compute the velocity components. The velocity vector is given by v = (dx/dt, dy/dt). Differentiate x(t) and y(t) with respect to t to find dx/dt and dy/dt. For x(t), differentiate (1/2 t^2 - 2t^2) with respect to t. For y(t), differentiate (1/2 t^2 - 2t) with respect to t.

Step 3: Evaluate the velocity components at t = 0 s and t = 4 s. Substitute t = 0 s and t = 4 s into the expressions for dx/dt and dy/dt to find the velocity components (vx, vy) at these times.

Step 4: Calculate the angle θ from the x-axis. The angle θ is given by θ = arctan(vy/vx). Use the velocity components (vx, vy) at t = 0 s and t = 4 s to compute θ for each time. Note that the arctan function accounts for the direction of motion based on the signs of vx and vy.

Step 5: Interpret the results. The angle θ at t = 0 s and t = 4 s represents the direction of motion of the particle relative to the x-axis at those specific times. Ensure the angle is expressed in degrees or radians as required by the problem.

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Key Concepts

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

Trajectory

Trajectory refers to the path that a particle follows as it moves through space over time. In this context, the trajectory is defined by the equations for x and y coordinates as functions of time (t). Understanding the trajectory is essential for analyzing the motion of the particle and determining its position at any given time.

Velocity

Velocity is a vector quantity that describes the rate of change of an object's position with respect to time. It has both magnitude and direction. To find the direction of motion at specific times, we need to calculate the velocity components by differentiating the position equations with respect to time, which will help us determine the angle of motion relative to the x-axis.

Angle of Motion

The angle of motion is the angle formed between the velocity vector of a particle and a reference axis, typically the x-axis. This angle can be calculated using the arctangent function, which relates the y-component of velocity to the x-component. Knowing the angle of motion at specific times allows us to understand how the particle is moving in relation to the coordinate system.

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Textbook Question

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