A fish swimming in a horizontal plane has velocity 𝐯i=(4.00...

A fish swimming in a horizontal plane has velocity $\mathbf{v}_{i}=(4.00 \mathbf{i}+1.00 \mathbf{j}) \mathrm{m} / \mathrm{s}$ at a point in the ocean whose displacement from a certain rock is $\mathbf{r}_{i}=(10.0 \mathbf{i}-4.00 \mathbf{j})$ $\mathrm{m}$. After the fish swims with constant acceleration for $20.0 \mathrm{~s}$, its velocity is $\mathbf{v}=(20.0 \mathbf{i}-5.00 \mathrm{j}) \mathrm{m} / \mathrm{s}$. (a) What are the components of the acceleration? (b) What is the direction of the acceleration with respect to the unit vector i? (c) Where is the fish at $t=25.0 \mathrm{~s}$ if it maintains its original acceleration and in what direction is it moving?

A fish swimming in a horizontal plane has velocity $\mathbf{v}_{i}=(4.00 \mathbf{i}+1.00 \mathbf{j}) \mathrm{m} / \mathrm{s}$ at a point in the ocean whose
displacement from a certain rock is $\mathbf{r}_{i}=(10.0 \mathbf{i}-4.00 \mathbf{j})$ $\mathrm{m}$. After the fish swims with constant acceleration for $20.0 \mathrm{~s}$, its velocity is $\mathbf{v}=(20.0 \mathbf{i}-5.00 \mathrm{j}) \mathrm{m} / \mathrm{s}$. (a) What
are the components of the acceleration? (b) What is the direction of the acceleration with respect to the unit vector i? (c) Where is the fish at $t=25.0 \mathrm{~s}$ if it maintains its original acceleration and in what direction is it moving?

Key Concepts

Component Analysis

Component analysis involves resolving vector quantities into their perpendicular components, typically along the x and y axes in planar motion. This technique simplifies the study of motion because it allows you to apply one-dimensional kinematic equations separately to each direction.

Vector Direction Determination

Determining the direction of a vector involves calculating the angle it makes with a reference axis, often using inverse trigonometric functions such as the arctan of the ratio of its components. This is essential for understanding the orientation of a vector in a coordinate system and for accurately describing its direction of action.

Vector Kinematics

Vector kinematics is the study of motion using vectors to represent displacement, velocity, and acceleration, each of which has both magnitude and direction. This concept allows us to describe and analyze motion in multiple dimensions by breaking down motion into its independent components along defined axes.

Constant Acceleration Equations

Constant acceleration equations are fundamental in kinematics when an object moves with a uniform acceleration. These equations relate initial velocity, final velocity, displacement, acceleration, and time, enabling the calculation of one quantity when the others are known, based on formulas such as v = v? + at and r = r? + v?t + 0.5at².

Transcript

00:01 Hello.

00:03 In this question, we have a fish that is moving in the sea, and the fish originally was at the position vector 10 in negative 4.

00:15 So the position initially was 10 in the eye direction, in negative 4 in the j direction with respect to a certain rock.

00:28 So assuming that the rock is at the origin, the fish was initially.

00:33 At 10 and negative 4, so the fish was almost here.

00:39 The fish initial velocity was 4 in the eye direction and 1 in the j direction.

00:51 After 20 seconds, the final velocity, so the velocity at 20, is 20 in the eye direction and negative 5 in the j direction.

01:07 So first we are asked to calculate the components of the acceleration.

01:14 Of course assuming that the acceleration was constant through the whole 20 seconds.

01:20 So in order to do this we will just use the equation of delta v by delta t.

01:30 So delta v is vf minus vi in vector notation divided by 20.

01:40 So that's basically 20, negative 5 minus 4 and 1 divided by 20.

01:55 And this gives us 0 .8 in the direction of i and 0 .3 in the j direction.

02:05 So those are the components of the acceleration.

02:08 The i component is 0 .8 and the j component is 0 .3.

02:17 And the second part, we're asked to get the direction of this acceleration with respect to the i.

02:26 So that's the i unit vector and that's the j unit vector.

02:33 The components are 0 .8 in the direction of i and 0 .3 in the direction of j.

02:41 So theta is the angle that the acceleration vector makes with the i direction, and this will be equal to, so theta will be equal to 10, inverse, 0 .3 divided by 0 .8, and this will give us 20 degrees and 33.

03:07 So that's the direction.

03:12 The acceleration vector makes an angle 20 and 33 with the i direction.

03:21 Finally, we're asked to know the position of the fish after 25 seconds, assuming that it was moving with the same acceleration in the same direction.

03:36 So let's first get the general formula for the position.

03:41 So now we are getting the position vector at any point in time.

03:48 This is basically our dot plus vi multiplied by t plus half 80 squared.

04:02 This is all vectors as well...

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