Physics, 28.12.2019 06:31 camila68

Aparticle leaves the origin with an initial velocity and a constant acceleration . when the particle reaches its maximum x coordinate, what are

(a) its velocity,
(b) its position vector?

Answers: 3

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Answers

Answer from: aria0826

(a)

$\displaystyle \vec v_f=$

(b)

$\displaystyle \vec r=$

Explanation:

Constant Acceleration Motion

This describes a situation where an object is constantly changing its velocity over time, at a constant rate. If we treat the magnitudes as scalars (one fixed direction) then we have the following expression to find the final speed v_f in term of the initial speed v_o , the constant acceleration a and the time t

v_f=v_o+at

The distance the object travels can also be found in terms of the same magnitudes

$\displaystyle x=v_ot+\frac{at^2}{2}$

If the object is moving in a two-dimension space, then all the magintudes are vectors, and the formulas change to

$\vec v_f=\vec v_o+\vec at$

$\displaystyle \vec r=\vec v_ot+\frac{\vec at^2}{2}$

The initial components of $\vec r$ are zero, as indicated in the question, so they are not included in the above formula.

The velocities and acceleration have two components in the x and y coordinates

The question does not clearly specify some conditions of the situation, so we'll assume some minimum facts to make this have sense

Assuming vx_o is positive, if the object will eventually reach a maximum x-coordinate, then the x-component of the acceleration must be negative, so the x-component of the velocity decreases to zero and the object moves back in that direction. The acceleration will be written as

$\vec a=$

Where ax is negative

The velocity can be written in vectorial notation as

$\vec v_f=$

Where vx_o and vy_o are the initial velocity components

We can find the time t_m where the object starts to change its direction in the x-axis by equating the v_f component to zero

vx_o+ax.t_m=0

Solving for t_m

$\displaystyle t_m=\frac{vx_f-vx_o}{a}=\frac{0-vx_o}{ax}$

$\displaystyle t_m=-\frac{vx_o}{ax}$

Since ax is negative, t_m is guaranteed to be positive.

(a)

The velocity can be computed at this specific time as

$\vec v_f=$

Replacing t_m

$\vec v_f=$

Simplifying

$\displaystyle \vec v_f=$

(b)

The postition vector at this specific time will be

$\displaystyle \vec r=\vec v_ot_m+\frac{\vec at_m^2}{2}$

$\displaystyle \vec r=.\left ( -\frac{vx_o}{a} \right )+\frac{.\left ( -\frac{vx_o}{a} \right )^2}{2}$

Operating

$\displaystyle \vec r=+$

Simplifying

$\displaystyle \vec r=$

Note: Assuming vx_o was positive doesn't make our formulas less valid. If vx_o was negative, then ax must be positive

Answer from: Quest

insects!

explanation:

Answer from: Quest

Alinear relationship would look like a straight line on the graph. it could have a positive negative or 0 slope. so it could be a line potentially going in any direction but it must be perfectly straight.

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Aparticle leaves the origin with an initial velocity and a constant acceleration . when the particle...

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