A proton initially has v⃗=4.0 î-2.0 ĵ+3.0 k̂ and then 4.0 s later has v⃗=-2.0 î-2.0 ĵ+5.0 k̂ (i

A proton initially has v⃗=4.0 î-2.0 ĵ+3.0 k̂ and then 4.0 s...

A proton initially has $\vec{v}=4.0 \hat{\mathrm{i}}-2.0 \hat{\mathrm{j}}+3.0 \hat{\mathrm{k}}$ and then $4.0 \mathrm{~s}$ later has $\vec{v}=-2.0 \hat{\mathrm{i}}-2.0 \hat{\mathrm{j}}+5.0 \hat{\mathrm{k}}$ (in meters per second). For that $4.0 \mathrm{~s}$, what are (a) the proton's average acceleration $\vec{a}_{\text {avg }}$ in unitvector notation, (b) the magnitude of $\vec{a}_{\text {avg }}$, and $($ c) the angle between $\vec{a}_{\text {avg }}$ and the positive direction of the $x$ axis?

A proton initially has $\vec{v}=4.0 \hat{\mathrm{i}}-2.0 \hat{\mathrm{j}}+3.0 \hat{\mathrm{k}}$ and then
$4.0 \mathrm{~s}$ later has $\vec{v}=-2.0 \hat{\mathrm{i}}-2.0 \hat{\mathrm{j}}+5.0 \hat{\mathrm{k}}$ (in meters per second). For
that $4.0 \mathrm{~s}$, what are (a) the proton's average acceleration $\vec{a}_{\text {avg }}$ in unitvector notation, (b) the magnitude of $\vec{a}_{\text {avg }}$, and $($ c) the angle between $\vec{a}_{\text {avg }}$ and the positive direction of the $x$ axis?

Key Concepts

Angle Calculation Between Vectors

Calculating the angle between vectors often involves using trigonometric functions, such as the cosine function. By relating the components of a vector to its magnitude, one can determine the angle a vector makes with respect to a coordinate axis. This technique is useful in decomposing vectors into components and understanding their directional relationships.

Magnitude of a Vector

The magnitude of a vector represents its size or length and is calculated by taking the square root of the sum of the squares of its components. This scalar value is essential for quantifying how large a vector is, independent of its direction.

Unit Vector Notation

Unit vector notation expresses any vector in terms of its directional components along an orthogonal basis, typically involving the i, j, and k unit vectors. This notation simplifies vector arithmetic by clearly indicating the direction associated with each component, making it easier to perform vector operations in physics and engineering problems.

Average Acceleration

Average acceleration is defined as the change in velocity divided by the time interval over which the change occurs. It is a vector quantity, meaning it has both magnitude and direction. This concept is fundamental in kinematics as it describes how quickly an object's velocity is changing on average over a given period.

Vector Subtraction

Vector subtraction is the process of finding the difference between two vectors by subtracting their respective components. This method is used to determine the change in velocity (or any other vector quantity) by subtracting the initial vector from the final vector, which is crucial for calculating quantities such as acceleration.

Transcript

00:01 In this problem we have two vectors, one from the initial and one for the final velocity.

00:06 And we know that four seconds passed between one stage and the other.

00:11 And first in part a we want to find the average acceleration.

00:15 So let's define what average acceleration really means.

00:21 That is the final velocity minus the initial velocity over the change in time.

00:31 So let's apply this equation to each of the component.

00:34 Of the vectors.

00:37 First, with the x components, you can say that the acceleration is going to be negative 2 minus 4 all over 4 seconds.

00:52 And that is going to give us negative 1 .5.

01:01 Then let's go to the y component.

01:08 There we have negative 2, negative times negative 2, that is going to be plus plus 2 over 4.

01:17 This gives us 0, and then we have k -hap, which is going to be equal to 5 minus 3 over 4.

01:38 And that is going to give us 0 .5.

01:43 Now we can write this in vector notation and say that the average acceleration is equal to negative 1 .5.

01:56 I had no y component and finally 0 .5 kha.

02:08 And let's remember that these two are in meters per second squared.

02:16 All right.

02:17 Now in part b we want to find the magnitude of the average acceleration.

02:24 And the magnitude of any vector is simply going to be the square root of its components squared and added...

Please give Ace some feedback