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

step 1 for this problem on the topic of motion in two dimensions. We are told that a fish that is swimm

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

A fish swimming in a horizontal plane has velocity $\overrightarrow{\mathbf{v}}_{i}=$ $(4.00 \hat{\mathbf{i}}+1.00 \hat{\mathbf{j}}) \mathrm{m} / \mathrm{s}$ at a point in the ocean where the= position relative to a certain rock is $\overrightarrow{\mathbf{r}}_{i}=(10.0 \hat{\mathbf{i}}-4.00 \hat{\mathbf{j}}) \mathrm{m}$ After the fish swims with constant acceleration for 20.0 $\mathrm{s}$ its velocity is $\overrightarrow{\mathbf{v}}=(20.0 \hat{\mathbf{i}}-5.00 \hat{\mathbf{j}}) \mathrm{m} / \mathrm{s} .$ (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

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.

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.

Transcript

00:01 For this problem on the topic of motion in two dimensions.

00:04 We are told that a fish that is swimming in a horizontal plane has a given initial velocity at a point in the ocean where its position is relative to a certain rock and that position is given.

00:16 The fish swims with constant acceleration for 20 seconds until it reaches a given final velocity and we want to calculate the components of the acceleration, the direction of the acceleration with respect to unit vector i and the direction that is moving if it maintains constant acceleration.

00:37 So firstly, the initial velocity v -i, we can see is 4 times unit vector i plus 1j, and that's meters per second.

00:57 Its final velocity is given as v, and v is equal to, and so this is v as a function of time, where time t is 20 seconds, and this is given as 20 i minus 5j.

01:24 And this again is in meters a second.

01:29 So firstly, we need to calculate the acceleration.

01:33 So the x component of acceleration, a x, is the rate of change of the x component of the fission's velocity, delta vx, delta t.

01:44 And this is 20 minus 4 looking at only the components along the i direction divided by the change in time, which is 20 seconds.

02:03 So our final answer here will be in meters per square second, which we know is the si unit for acceleration, and we get this to be 0 .8 meters per square second.

02:19 Similarly, the y component of the acceleration is delta v y by delta t.

02:28 And so this is minus 5 minus 1 over the change in time is again 20, and that's in meters per square second, which gives us the y component of acceleration to be minus 0 .3 meters per square second.

02:52 So we found the acceleration of the fish at time t is equal to 20 seconds.

03:03 For the next part of the problem, we want to find the direction of the acceleration with respect to unit vector i...

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