Heather in her Corvette accelerates at the rate of (8.00...

Key Concepts

Constant Acceleration Kinematics

This concept involves the equations that govern motion under unchanging acceleration. In such scenarios, an object's displacement and final velocity after a given time can be calculated from initial conditions and acceleration. This is fundamental in solving problems where objects start from rest or have uniform acceleration, as it allows for determining positions, speeds, and distances traveled.

Relative Motion

Relative motion analyzes the behavior of one moving object from the reference frame of another. It involves determining differences in displacement, velocity, or acceleration by subtracting one object's vector quantities from another’s. This concept is crucial for understanding how different observers perceive the motion of objects differently, especially when they are both in motion.

Vector Operations

Vector operations, such as addition, subtraction, and calculating magnitudes, are essential for handling physical quantities that have both magnitude and direction. In mechanics, representing acceleration, velocity, and displacement as vectors and performing operations on them allows for accurate analysis of motion in two or three dimensions.

Magnitude of a Vector

Calculating the magnitude of a vector involves using the Pythagorean theorem to combine its components into a single scalar value, such as speed or distance. This method is fundamental when converting a two-dimensional vector into a usable measure of separation or relative speed.

Transcript

00:01 Hello.

00:03 In this problem we have two persons that are each in her car and both are starting from rest and at the same position from the origin with hether accelerating according to this vector equation.

00:22 3 in the i direction minus 2 in the j direction and this is of course measured in meters per set.

00:31 Square, while gel is moving with another acceleration that is one in the i direction and plus 3 in the j direction.

00:48 Again, that's in meters per second square.

00:51 The first thing, we're asked to calculate the average, the relative velocity of heather with respect to gel.

01:02 So that's the first part, we need to know the heather's speed with respect to gel.

01:09 So in order to do this, let's first define the velocity of each one of them.

01:15 So now we have the velocity of heather.

01:20 From here, just integrating, we know that we get the velocity, given that they started from rest and they started from the origin, so we don't have to care about the constants.

01:30 So so basically integrating the acceleration of heather, we have that 3t in the eye direction, minus 2t in the j direction, and all of this is measured in meters per second.

01:47 And now we have the velocity of gel.

01:51 Again, integrating, we get t in the r direction, and we have plus 3t in the j direction.

01:59 Now we need to know the velocity of heather with respect to gel.

02:07 So straightforward, this is just the velocity of heather minus the velocity of gel.

02:15 Okay, so subtracting those two vectors from each other to give a general expression for the relative velocity in terms of time.

02:25 So now subtracting this, we get 2t in the eye direction, and we get minus 2, minus 3, so that's minus 5.

02:39 T and the j's direction in meters per second.

02:44 So first, now we're asking a.

02:49 The relative velocity of heather with respect to gel after 5 seconds.

02:54 So now we need to get the relative velocity of heather with respect to gel at 5.

03:00 Just substituting here for t equals 5.

03:04 And we get the answer of 10 in the eye direction minus 25 in the j direction and all of this is in meters per second.

03:21 So that's the answer of part a.

03:23 Let's box this for your reference first.

03:27 And then let's move for part b.

03:32 In part b, we're asked to know how far exactly are they from each.

03:38 Other so let's assume that this is the origin and each one of them started to move in a certain direction with heather moving in the positive x -axis and the negative y -axis so it started to move in this direction and gel started to move in the positive x -axis and the positive y -axis as well so it started to move in kind of this direction so i how far they are apart can be represented by this vector pointing from one's position to the other's position.

04:17 So first, let's get the position vector of each one of them.

04:21 So now we need to get the r of heather.

04:25 And according to the kinematics equation, this is basically the v.

04:30 T plus half a t squared.

04:35 And here we know that the initial velocity was zero for each...

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