Useful background for this problem can be found in Multiple-...

Useful background for this problem can be found in Multiple- Concept Example 2 . On a spacecraft two engines fire for a time of 565 . One gives the craft an acceleration in the $x$ direction of $a_{x}=5.10 {m} / {s}^{2}$ while the other produces an acceleration in the $y$ direction of $a_{y}=7.30 {m} / {s}^{2}$ At the end of the firing period, the craft has velocity components of $v_{x}=3775 {m} / {s}$ and $v_{y}=4816 {m} / {s} .$ Find the magnitude and direction of the initial velocity. Express the direction as an angle with respect to the $+x$ axis.

Useful background for this problem can be found in Multiple- Concept Example 2 . On a spacecraft two engines fire for a time of 565 . One gives the craft an acceleration in the $x$ direction of $a_{x}=5.10 {m} / {s}^{2}$ while the other produces an acceleration in the $y$ direction of $a_{y}=7.30 {m} / {s}^{2}$
At the end of the firing period, the craft has velocity components of $v_{x}=3775 {m} / {s}$ and $v_{y}=4816 {m} / {s} .$ Find the magnitude and direction of the initial velocity. Express the direction as an angle with respect to the $+x$ axis.

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Key Concepts

Constant Acceleration Kinematics

This concept involves using the kinematic equations for constant acceleration to relate initial velocity, acceleration, time, and final velocity. The basic equation, v = v0 + at, applies separately in each component direction, allowing determination of the unknown initial velocity components when the acceleration and final velocity are provided.

Vector Decomposition

This concept is the process of breaking down a vector into its component parts along perpendicular axes, typically the x and y axes. Decomposition allows for solving multidimensional motion problems by analyzing each dimension independently using the corresponding kinematic equations.

Vector Magnitude Calculation

This concept refers to finding the total magnitude of a vector using its components, typically by applying the Pythagorean theorem. When the x and y components of a velocity are known, the overall speed is calculated as the square root of the sum of the squares of the individual components.

Determination of Vector Direction

This concept involves calculating the angle that a vector makes with a reference axis, generally using the inverse tangent function. By taking the ratio of the y-component to the x-component of the vector, the direction can be determined as an angle relative to the positive x-axis.

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