A cyclist travels from point A to point B in 10 min. During...

A cyclist travels from point A to point B in 10 min. During the first 2.0 min of her trip, she maintains a uniform acceleration of $0.090 \mathrm{m} / \mathrm{s}^{2} .$ She then travels at constant velocity for the next 5.0 min. Next, she decelerates at a constant rate so that she comes to a rest at point $\mathrm{B}$ 3.0 $\mathrm{min}$ later. (a) Sketch the velocity-versus-time graph for the trip. (b) What is the acceleration during the last 3 min? (c) How far does the cyclist travel?

A cyclist travels from point A to point B in 10 min. During the first 2.0 min of her trip, she maintains a uniform acceleration of $0.090 \mathrm{m} / \mathrm{s}^{2} .$ She then travels at constant
velocity for the next 5.0 min. Next, she decelerates at a constant rate so that she comes to a rest at point $\mathrm{B}$ 3.0 $\mathrm{min}$ later. (a) Sketch the velocity-versus-time graph for the trip.
(b) What is the acceleration during the last 3 min? (c) How far does the cyclist travel?

Key Concepts

Piecewise Motion

Piecewise motion involves breaking the motion into segments with distinct characteristics, such as acceleration, constant velocity, and deceleration. This approach allows for separately evaluating and then summing the contributions to total displacement.

Kinematics Equations

These equations relate displacement, velocity, acceleration, and time and are fundamental to solving motion problems in physics. They are applied separately to the different segments of motion in a piecewise analysis.

Velocity-Time Graph

A velocity-time graph visually represents an object’s speed over time, with different slopes indicating periods of acceleration, constant velocity, or deceleration. It is used to understand and analyze changes in motion and to calculate displacement via the area under the curve.

Constant Velocity

This occurs when an object moves at a fixed speed, meaning its acceleration is zero. It simplifies displacement calculations to a simple multiplication of speed and time, which is crucial for the middle segment of piecewise motion.

Uniform Acceleration

This concept describes motion where an object's velocity changes at a constant rate over time. It is essential for calculating the final speed and displacement during the acceleration phase using kinematic equations.

Deceleration

Deceleration refers to negative acceleration, where the object reduces its speed at a constant rate. It is analyzed using the same kinematic formulas as acceleration, with attention to the sign change, to determine the time to rest and distance covered.

Transcript

00:01 So this problem is a combination of a kinematics and graphing problem.

00:05 You're told you have a cyclist going from point a to point b, and it takes 10 minutes, but the problem broke it up into three individual parts.

00:13 So for the first part, you're told from zero to two minutes, so that's the amount of time for that segment, i guess delta t, that they have an acceleration a of 0 .090 meters.

00:33 Per second squared.

00:35 You're told, you have to assume in this problem, it doesn't say it.

00:39 This could be an issue if you didn't assume it.

00:41 I don't think you can solve it.

00:42 Otherwise, that the rider starts from rest.

00:45 So that may be a flaw in this problem.

00:46 They need to include that information.

00:48 So that you're told for part one, for part two of the problem, you're told that the delta t is from two minutes up to seven minutes, and that for this period, the acceleration is zero.

01:05 They're moving with a constant velocity.

01:08 And for part three of the problem, delta t of seven minutes to 10 minutes, you have a final velocity that cyclist comes to rest of zero.

01:23 You're told, you know the total time is three minutes.

01:27 And so you need to solve this problem, giving this information.

01:31 The first thing they ask for is that you sketch a graph of velocity as a function of time.

01:37 And so v is a function of t and you're told that it's moving forward the whole time it ends up at rest and so out here at 10 seconds at 10 minutes the object going to be on the x -axis at zero working your way backwards let's see if this is 5 6 7 8 9 10 and 0 1 2 3 4 5 so at 7 which is here they were at some velocity, which we don't know, and that velocity heads down to zero.

02:16 So it's decelerating, not really a great term, but it's accelerating opposite the direction of velocity.

02:21 We're told at two minutes, it started moving with a constant velocity.

02:25 And so the velocity is constant for this segment.

02:28 And assuming it started from rest, as we've said, it's going to go up like this for the first two minutes.

02:34 So that's kind of what the graph of velocity versus time would look like, and we can add some information to just a moment.

02:40 So for part one, we're know enough with via and t to get vf, the final velocity for this segment using vf equals vi plus at...

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