A fish swimming in a horizontal plane has velocity 𝐯i=(4.00 𝐢̂+1.00 𝐣̂) m / s at a point in the

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

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

Key Concepts

Vector Representation and Components

In physics, vectors such as velocity, acceleration, and displacement are defined by both magnitude and direction. They are typically expressed in terms of components along unit vectors (like i? and j?) which allows separate analysis in orthogonal directions.

Constant Acceleration Kinematics

When an object moves with constant acceleration, its motion can be described with kinematic equations that relate displacement, initial velocity, acceleration, and time. This framework enables predictions of future position and velocity based on initial conditions and uniform acceleration.

Acceleration as Change in Velocity

Acceleration measures the rate at which velocity changes over time. In vector form, it is computed by subtracting the initial velocity vector from the final velocity vector and dividing the result by the elapsed time, with the operation performed component-wise.

Determining the Direction of a Vector

The direction of a vector, often given as an angle relative to a reference axis like i?, is determined by calculating the arctangent of the ratio of its components. Expressing directions with unit vectors standardizes the orientation in the coordinate system.

Transcript

00:01 This question we are given a fish, it has initial velocity for i plus j meters per second and initial position 10 i minus 4 j meters.

00:12 After 20 seconds, the final velocity of the fish is 20 i minus 5 j meters per second.

00:18 So there are three parts in this question.

00:21 In part a, want to find the acceleration.

00:23 Part b, want to find the direction of acceleration relative to the ihead direction.

00:29 And press c when to find a final velocity of the fish at t go to 25 seconds and the final position of the fish at t goes to 25 seconds.

00:42 So in prime a, okay, we are given that a is delta v over delta t because the acceleration is constant so we are, we can use this formula.

01:01 So the t is v final minus v initial then divide by delta t so we final is 20 i had minus 5 j heads and then the initial velocity is for i plus j okay and then divide by 20 if you do this you get 0 .8 i had minus 0 .3 j head head okay okay basically is just grouping the terms so for the i had is 20 minus 4 and then you divide by 20 you get 0 .8 and then for the j components it's minus 5 minus 1 so minus 6 divide by 20 you get minus 0 .3 okay and then the units are meters per second square okay so this will be the answer for part a okay then in part b you want to find a direction of acceleration relative to the i component so so the i component is pointing this way the acceleration is pointing something like this in this direction so the angle that the question is referring to is this big angle okay i will first find data okay data is easier to find so uh tangent data is the y component divide by the x component 0 .3 divide by 0 .8 this is uh are 0 .375.

02:49 So the angle would be 20 .6 degrees.

02:55 And then the direction of a, relative to ihat is equal to 2060 degrees minus data.

03:13 So we get 339 degrees.

03:17 Okay.

03:25 So this is the answer for part b...

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