Two soccer players, Mary and Jane, begin running from nearly...

Key Concepts

Law of Cosines

The law of cosines is used to relate the lengths of sides in a triangle to the cosine of one of its angles. This method is particularly useful when determining the distance between objects moving in different directions when their paths and the angle between them are known.

Kinematics in Two Dimensions

Kinematics in two dimensions deals with the motion of objects in a plane where both horizontal and vertical components must be considered. It involves determining quantities like displacement, velocity, and time by applying principles of uniform motion and vector addition or subtraction.

Relative Velocity

Relative velocity is the velocity of one object as observed from another moving object. It is determined by subtracting one object's velocity vector from the other's. This concept is crucial in analyzing the motion of objects relative to each other, such as calculating how fast the distance between them changes over time.

Vector Decomposition

Vector decomposition involves breaking a vector into its horizontal and vertical components. This technique is essential when dealing with two-dimensional motion, as it allows the separation of motion into perpendicular directions, simplifying calculations and analysis.

Transcript

00:01 Okay, so for this question in the first part it's asking how long is it before they are 25 meters apart so that we'll do part a so we need to find the time it takes for mary and jane to get 25 meters apart so i've drawn a little diagram up here in the right where mary's running in an easterly direction at four meters per second and jane is going in a northeastern direction at 5 .4 meters per second and jane is going in a northeastern direction at 5 .4 meters per second and so what we're going to need here is we're going to need the x and y components of the velocity of jane, i think it is, who's going northeast.

00:56 So we're going to need this y component and we're going to need this x component.

01:04 So if we plug that into our calculator, we'll get 5 .4 times sine of 60 for the opposite, which is about 4 .4 .4 .5 .5 .5 .5 .5 .5 .5 .5 .5 .4 .4.

01:13 For the opposite, which is about 4 .4.

01:16 So that's here, 4 .68 in the x direction.

01:21 And then in the y direction, we'll get 5 .4 times cosine of 60, which is 2 .7.

01:30 So those are our, are those velocities we need.

01:35 So for part a, if we're going to find the time it takes, we're going to need to find a way to get this distance between them, which is going to be kind of difficult because mary is going to this way.

01:50 And jane's going this way and so mary's going to be moving along here and maybe she'll be here and jane will be going along here even faster so might be here but we need this distance here.

02:02 So for this distance we'll have an x component right here and a y component and we'll need to use pythagorean theorem to get the hypotenuse here which is our d.

02:15 So if we use pythagorean theorem, we'll get d equals the square root of dx squared plus dy squared.

02:29 So let's start plugging in some of the things we know.

02:33 So our dx squared is going to be equal to the distance of mary from the origin minus distance of jane in the x direction.

02:52 So we'll just call that djx.

02:56 So that'll be our distance in the y direction between the two.

03:00 And then our distance in the y direction will be the distance of jane going in the y direction.

03:05 Because mary isn't going up or down.

03:07 She's just going to the right.

03:09 So we'll do d of j, y, squared.

03:14 Okay.

03:16 So we have our equation here where velocity is equal to distance over time.

03:21 Distance is velocity times time.

03:23 So we can plug those things in and we'll then we'll solve for time.

03:30 So let's plug that in.

03:32 So we'll get this bit velocity of mary times time minus the velocity in the x direction of jane times time squared plus velocity of jane in the y direction times time.

04:01 All right.

04:02 So we have all of these numbers actually.

04:03 We already got them over here.

04:05 So we'll need this one, this one, and this one.

04:10 So let's plug those in as well.

04:11 I'll draw it a little bit bigger because we're going to have lots of numbers in here.

04:16 So the velocity of mary's 4 times time minus the velocity of jane in the x direction, which is 2 .7 times time squared plus the velocity of jane in the y direction, which is 0 .4 .68 times time and squared.

04:47 So we're going to have to do a bit of algebra here and to make this work.

04:58 So let's do that.

05:03 So we can get time on its own.

05:08 That's going to take a little bit.

05:09 So we got 4t times 4t minus 4t times 2 .70 minus 2 .70 minus 2 .70 times 2 .70 times 2 .70 times 4t plus 2 .7 t times.....

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