A record of travel along a straight path is as follows: 1....

A record of travel along a straight path is as follows: 1. Start from rest with a constant acceleration of $2.77 \mathrm{~m} / \mathrm{s}^{2}$ for $15.0 \mathrm{~s}$ 2. Maintain a constant velocity for the next $2.05 \mathrm{~min}$. 3. Apply a constant negative acceleration of $-9.47 \mathrm{~m} / \mathrm{s}^{2}$ for $4.39 \mathrm{~s}$. (a) What was the total displacement for the trip? (b) What were the average speeds for legs 1,2, and 3 of the trip, as well as for the complete trip?

A record of travel along a straight path is as follows:
1. Start from rest with a constant acceleration of $2.77 \mathrm{~m} / \mathrm{s}^{2}$ for $15.0 \mathrm{~s}$
2. Maintain a constant velocity for the next $2.05 \mathrm{~min}$.
3. Apply a constant negative acceleration of $-9.47 \mathrm{~m} / \mathrm{s}^{2}$ for $4.39 \mathrm{~s}$.
(a) What was the total displacement for the trip? (b) What were the average speeds for legs 1,2, and 3 of the trip, as well as for the complete trip?

Key Concepts

Constant Acceleration

This concept involves motion in which an object's velocity changes at a steady rate. In problems involving constant acceleration, kinematic equations are used to determine various aspects of the motion such as final velocity, displacement, and time. It is crucial when analyzing parts of a journey where the acceleration is not zero.

Constant Velocity

Constant velocity refers to motion where an object's speed and direction remain unchanged. In such scenarios, displacement can be directly calculated by multiplying speed and time, as there is no acceleration affecting the movement.

Kinematic Equations

These are mathematical formulas that relate displacement, initial velocity, final velocity, acceleration, and time for objects in linear motion. They provide the basis for solving problems involving both constant acceleration and constant velocity, and are essential tools in the study of classical mechanics.

Displacement

Displacement is the overall change in position of an object. In multi-phase motion problems, it is determined by summing the displacements from each segment of the motion, taking into account the nature of the motion in each phase (accelerated or constant velocity).

Average Speed Analysis

Average speed is defined as the total distance traveled divided by the total time taken. Analyzing average speeds over different segments of a journey, as well as for the entire trip, provides insights into performance during different phases of motion and helps in understanding combined effects of acceleration and constant velocity.

Transcript

0:00 All right.

00:02 So this is travel along a path.

00:06 You start from rest with an acceleration, an acceleration of 2 .77 meters per second squared for 15 seconds.

00:21 Okay, then constant velocity for 2 .05 minutes.

00:28 So, v2 for t equals 2 .05 minutes, but there's 60 seconds in a minute.

00:47 Then negative acceleration, 9 .47 meters per second squared for 4 .39 seconds.

01:03 So what was the total displacement of the trip? so x equals x initial, which is zero, plus v initial t, which is also zero, plus one -half a -t squared.

01:24 But we're also going to need the velocity.

01:27 V is v -0, which is zero, plus a -t.

01:31 And so i'm going to call this v2 and i'm going to call this x1.

01:38 So putting these into the desmos graph and calculator, i get a, i need to put an a is 2 .77 and t is 15.

01:52 So x1 is one half a t squared, which is going to be 312 meters.

02:13 V2 is at, which is going to be 41 .6 meters.

02:25 So if you travel at this velocity for that time, i'm going to call this t2.

02:33 So x2 is going to be v2 times t2.

02:40 So putting that into the calculator, x2 equals v2, t2, v2, v2, t2.

02:54 Aha.

02:55 T2, i didn't put that in here yet, t2 is 2 .05 times 60.

03:07 So therefore, x2 is going to be 5 ,110 meters.