A particle starts from the origin at t=0 with a velocity of v⃗1=(8.0 m / s) ĵ and moves in the

A particle starts from the origin at t=0 with a velocity of...

A particle starts from the origin at $t=0$ with a velocity of $\vec{v}_{1}=(8.0 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{j}}$ and moves in the $x y$ plane with a constant acceleration of $\vec{a}=\left(4.0 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{i}}+\left(2.0 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{j}}$. At the instant the particle's $x$ coordinate is $29 \mathrm{~m}$, what are (a) its $y$ coordinate and (b) its speed?

A particle starts from the origin at $t=0$ with a velocity of $\vec{v}_{1}=(8.0 \mathrm{~m} / \mathrm{s}) \hat{\mathrm{j}}$ and moves in the $x y$ plane with a constant acceleration of $\vec{a}=\left(4.0 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{i}}+\left(2.0 \mathrm{~m} / \mathrm{s}^{2}\right) \hat{\mathrm{j}}$. At the
instant the particle's $x$ coordinate is $29 \mathrm{~m}$, what are (a) its $y$ coordinate and (b) its speed?

Key Concepts

Constant Acceleration Kinematics

This concept deals with the use of kinematic equations that describe the motion of an object with constant acceleration. These equations relate displacement, initial velocity, acceleration, and time, and are essential for analyzing motions in any direction. The fundamental equations help determine positions and velocities at any instant.

Component Analysis in Two-Dimensional Motion

In problems involving motion in a plane, the motion is often broken down into its independent x and y components. Each component can be analyzed separately using the kinematic equations, as the acceleration, velocity, and displacement in one direction do not affect those in the perpendicular direction. This decomposition simplifies the problem and allows for the calculation of time and displacement in each coordinate direction.

Time Determination from One Component

When one coordinate's displacement (or a related quantity) is known, the corresponding kinematic equation can be used to solve for the time taken to reach that displacement. Once the time is found using, for example, the x-component motion formula, it can be applied universally to all components due to the shared time across the motion.

Speed Calculation as the Magnitude of a Velocity Vector

Speed is a scalar quantity obtained as the magnitude of the velocity vector. It is calculated by taking the square root of the sum of the squares of the individual velocity components. This method of combining components ensures that the overall effect of motion in multiple dimensions is captured in a single, scalar speed value.

Transcript

00:01 All right, the particle starts at the origin and has a velocity of 8 in the j direction meters per second.

00:17 And its acceleration is 4 in the i direction plus 2 in the j direction meters per second.

00:41 So i'm trying to figure out, i'm going to do time first, when the x coordinate is 29.

00:53 So x is going to equal x initial, which is zero, plus vx initial, which is 8 times t, plus one -half times the acceleration, which is 4 t squared.

01:13 So x needs to be 29.

01:19 So i'm going to rewrite this as 2t squared plus 8t minus 29 equals 0.

01:36 And now i'm just going to graph it and look for the y intercept.

01:40 2t squared plus 8t minus 29.

01:51 So i'm graphing it, and i see that the x intercept is at 2 .30 seconds.

02:07 So now i'm going to write t equals 2 .30 in my calculator.

02:14 Okay.

02:16 The question now is, in 2 .3 seconds, what is its y coordinate? it.

02:27 Well, y is going to equal y initial.

02:32 Oh, no, that eight, that eight was in the, in the y direction.

02:41 E, e, e, e, e, so x is going to equal vx, which was zero, plus one half a t squared.

02:55 Oh, man.