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(III) The acceleration of a particle is given by $a=A \sqrt{t}$where $A=2.0 \mathrm{m} / \mathrm{s}^{5 / 2}$ . At $t=0, v=7.5 \mathrm{m} / \mathrm{s}$ and $x=0 .$(a) What is the speed as a function of time? (b) What is thedisplacement as a function of time? (c) What are the accel-eration, speed and displacement at $t=5.0 \mathrm{s} ?$

a) $v=7.5 m / s+\frac{2}{3}\left(2.0 m / s^{5 / 2}\right) t^{3 / 2}$b) $x=(7.5 m / s) t+\frac{4}{15}\left(2.0 m/ s^{5 / 2}\right) t^{5 / 2}$c) $a_{(t=5.0 s)}=\left(2.0 m/ s^{5 / 2}\right) \sqrt{5.0 s}=4.5 m / s^{2}$ $v_{(t=5.0 s)}=7.5 m / s+\frac{2}{3}\left(2.0 m / s^{5 / 2}\right)(5.0 s)^{3 / 2}=22.41 m/ s \approx 22 m / s$ $x_{(t=5.0 s)}=(7.5 m / s)(5.0 s)+\frac{4}{15}\left(2.0 m / s^{5 / 2}\right)(5.0 s)^{5 / 2}=67.31 m \approx 67 m$

Physics 101 Mechanics

Chapter 2

Describing Motion: Kinematics in One Dimension

Physics Basics

Motion Along a Straight Line

Motion in 2d or 3d

Newton's Laws of Motion

Cornell University

University of Michigan - Ann Arbor

Simon Fraser University

University of Sheffield

Lectures

03:28

Newton's Laws of Motion are three physical laws that, laid the foundation for classical mechanics. They describe the relationship between a body and the forces acting upon it, and its motion in response to those forces. These three laws have been expressed in several ways, over nearly three centuries, and can be summarised as follows: In his 1687 "Philosophiæ Naturalis Principia Mathematica" ("Mathematical Principles of Natural Philosophy"), Isaac Newton set out three laws of motion. The first law defines the force F, the second law defines the mass m, and the third law defines the acceleration a. The first law states that if the net force acting upon a body is zero, its velocity will not change; the second law states that the acceleration of a body is proportional to the net force acting upon it, and the third law states that for every action there is an equal and opposite reaction.

04:16

In mathematics, a proof is a sequence of statements given to explain how a conclusion is derived from premises known or assumed to be true. The proof attempts to demonstrate that the conclusion is a logical consequence of the premises, and is one of the most important goals of mathematics.

06:39

(III) The acceleration of …

03:36

A particle moves along a l…

01:59

The position of a particle…

01:57

A particle has a constant …

03:34

02:23

A particle moves in a stra…

03:25

(II) At $t=0,$ a particle …

03:48

Find the velocity, acceler…

08:04

A particle starts from the…

we're giving Ah, an acceleration, which is a rooty and a And this problem is just too. So we'll be right that. And we also know some initial conditions Where when t equals zero R velocity is seven point five and our position is zero. And so, in order to get our velocity, we know that from the definition of exhilaration, it's DVD t. So we can say that, uh, the integral Devi is gonna be the integral of a d t. And the left side. We get velocity that we have to integrate our acceleration, which is two point. Oh, and, uh, I'm gonna write square root of t as t to the 1/2 just to make it easier to use our power rule and e t. So now we just integrate. So this is gonna be 2.0, it's just a constant t to the 1/2. He increased the 1/2 to 3 halves and then divide by three have so when multiplied by 2/3 and we need to plus c here. So our velocity as a function of time is 4/3 t to the three halves plus C. And this is all possible velocities. But since we know that the initial velocity AT T equals zero, once we plug in zero for time, this is gonna equal 7.5 meters per second. And this just is zero. So R C is 7.5. So are one velocity function that fits all these criteria is 7.5 plus 4/3 t 23 halves. Now, if you want position, it's the same thing we just did. We know that velocity is the X DT and so we can integrate the X and integrates, uh, c b d t. So our position, the function of time is the integral of what we got up here. 7.5 plus 4/3 treated three halves ET. Now we just integrate So 7.5 ad t and 4/3 times t the three have. So we integrate and add 123 house we can t the five halves and then we have to divide by five. Have some multiply by 2/5 So and instead of see because we used to see in the last part of just confused de, it's a different constant, but it doesn't matter. And so we know our position function. It's gonna look like this. It's gonna be 8/15 sti to the five halves plus de and up here. We know that our position of Time T equals zero is zero So solve for D using that 7.5 times zero and this is all gonna equals zero. Now, this term goes to zero. This term ghost is us. That must mean that D equals zero. So our position as a function of time 7.5 t plus 8/15 3 to 5 halves. Now we want to know all three of these are position, velocity and acceleration at Time T equals five, so I'll just go through. All these functions are a is going to be to Times Square e five, and that gives us about 4.5 meters per second squared. And our velocity, uh, at five is 7.5 plus or thirds times five to the three halves, and that gives us about 22.4 meters per second. And finally, for a position 7.5 times five plus 8/15 times five. That five halves into our position at time Chief was five is around 67.3 meters

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