For the instant represented, car A has an acceleration in...

For the instant represented, car $A$ has an acceleration in the direction of its motion, and car $B$ has a speed of $45 \mathrm{mi} / \mathrm{hr}$ which is increasing. If the acceleration of $B$ as observed from $A$ is zero for this in stant, determine the acceleration of $A$ and the rate at which the speed of $B$ is changing.

Key Concepts

Instantaneous Kinematics

Instantaneous kinematics involves the study of an object's velocity and acceleration at a specific moment in time. This understanding is important for analyzing situations where instantaneous conditions, like velocity or acceleration, change and determining the consequent rates of change in the motion.

Relative Acceleration

This concept deals with the acceleration of one body as observed from another moving body. It is calculated as the difference between the absolute accelerations of the two bodies and is essential in understanding how motion is perceived differently from various reference frames.

Frame of Reference

A frame of reference is a coordinate system used to measure the position, velocity, and acceleration of an object. Analyzing motion from different reference frames can simplify problems, particularly when transforming between non-inertial and inertial frames where fictitious forces may come into play.

Vector Nature of Acceleration

Acceleration is a vector quantity, which means it has both magnitude and direction. Understanding this vector nature allows one to decompose the acceleration into components, such as tangential and normal, which is crucial in analyzing motion along curved trajectories or in different directions.

Transcript

00:01 For the instant represented in this diagram, car a has an acceleration in the direction of its motion, which i've listed there, and car b has the speed of 45 miles per hour, increasing.

00:14 If the acceleration of b is observed as zero at this instant, determine the acceleration a and the rate at which the speed of b is changing.

00:24 Okay, let's see if we can make any heads or tails out of this one.

00:28 Let's look for the acceleration of car b.

00:31 The general expression and i'm just going to go acc for acceleration of carb.

00:58 Okay.

01:20 When i have all these dots and little things, i have to be relatively quiet.

01:34 And then, okay, motion to b is circular.

01:48 I switch colors just so i can have different reference points.

01:58 So, and, and then we can find, so this times this equals vb.

02:39 Equals 45 miles per hour and that will equal 5 ,200 feet and 3 ,600 seconds.

03:10 This will equal 66 feet per second.

03:18 And with pb equal to 600 feet.

03:40 Then we'll get 600.

03:55 That will equal 0 .11 radiance per second.

04:08 Okay.

04:12 And the acceleration, okay, okay, the first term is a centripetal acceleration.

04:48 Second term is the speed of increase.

04:50 These coordinates are not constant, so we have to put them into cartesian.

04:54 So, okay, so this will be equal to the cosine plus the sign, and this will be equal to negative times the sign plus the cosine, plus the cosine, and that gives our acceleration for b.

05:32 So r, and then we'll plus.

06:18 Okay, cara accelerated linear.

06:21 I'm going to switch colors again...

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