A spaceship maneuvering near Planet Zeta is located at r=(600i−400j+200k)×103 km,\mathbf{r} = (600\mathbf{i} - 400\mathbf{j} + 200\mathbf{k}) \times $$10^3$$ \, \text{km}, relative to the planet, and traveling at v=9500i m/s\mathbf{v} = 9500\mathbf{i} \, \text{m/s}. It turns on its thruster engine and accelerates with a=(40i−20k) m/s2\mathbf{a} = (40\mathbf{i} - 20\mathbf{k}) \, \text{m/s}^2 for 35 min35\text{ min}. What is the spaceship's position when the engine shuts off? Give your answer as a position vector measured in km⁡\operatorname{km}.

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 44

Chapter 4, Problem 44

A spaceship maneuvering near Planet Zeta is located at $r = (600 i - 400 j + 200 k) \times 10^{3} km,$ relative to the planet, and traveling at $v equals 9500 i m slash s$ . It turns on its thruster engine and accelerates with $a = (40 i - 20 k) {m/s}^{2}$ for $35 min$ . What is the spaceship's position when the engine shuts off? Give your answer as a position vector measured in $km$ .

Verified step by step guidance

Step 1: Convert the time duration from minutes to seconds. Since 1 minute equals 60 seconds, multiply 35 minutes by 60 to get the time in seconds.

Step 2: Use the kinematic equation for position:

r_f = r_i + v_i t + (1/2) a t²

. Here,

r_i

is the initial position vector,

v_i

is the initial velocity vector,

a

is the acceleration vector, and

t

is the time duration.

Step 3: Substitute the given values into the equation. The initial position vector is

r_i = (600î - 400ĵ + 200k) × 10³

km, the initial velocity vector is

v_i = 9500î

m/s, and the acceleration vector is

a = (40î - 20k)

m/s². Ensure all units are consistent, converting the position vector to meters by multiplying by

10³

Step 4: Calculate each term in the equation. First, compute

v_i t

by multiplying the velocity vector by the time duration. Then compute

(1/2) a t²

by squaring the time duration, multiplying by the acceleration vector, and dividing by 2.

Step 5: Add the three vectors (

r_i

v_i t

, and

(1/2) a t²

) to find the final position vector. Convert the result back to kilometers by dividing by

10³

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Key Concepts

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

Position Vector

A position vector describes the location of a point in space relative to a reference point, typically the origin of a coordinate system. In this context, the spaceship's position vector is given in kilometers and indicates its distance and direction from Planet Zeta. The components of the vector represent the distances along the x, y, and z axes, allowing for a three-dimensional representation of the spaceship's location.

Acceleration

Acceleration is the rate of change of velocity of an object over time. It is a vector quantity, meaning it has both magnitude and direction. In this problem, the spaceship accelerates due to its thruster engine, which affects its velocity and ultimately its position. Understanding how to apply acceleration over a given time period is crucial for determining the final position of the spaceship after the engine shuts off.

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration. These equations relate displacement, initial velocity, final velocity, acceleration, and time. In this scenario, they are essential for calculating the spaceship's change in position due to its acceleration over a specified time interval. By applying these equations, one can determine the new position vector after the acceleration phase.

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