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?

(a)

(b)

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 in term of the initial speed , the constant acceleration a and the time t

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

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

The initial components of 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 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

Where ax is negative

The velocity can be written in vectorial notation as

Where and are the initial velocity components

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

Solving for

Since ax is negative, is guaranteed to be positive.

(a)

The velocity can be computed at this specific time as

Replacing

Simplifying

(b)

The postition vector at this specific time will be

Operating

Simplifying

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

where are the vectors ?

d eras and periods