A block of mass 2.00 kg is initially at rest at x=0 on a...

A block of mass $2.00 \mathrm{~kg}$ is initially at rest at $x=0$ on a slippery horizontal surface for which there is no friction. Starting at time $t=0,$ a horizontal force $F_{x}(t)=\beta-\alpha t$ is applied to the block, where $\alpha=6.00 \mathrm{~N} / \mathrm{s}$ anwd $\beta=4.00 \mathrm{~N}$. (a) What is the largest positive value of $x$ reached by the block? How long does it take the block to reach this point, starting from $t=0,$ and what is the magnitude of the force when the block is at this value of $x ?$ (b) How long from $t=0$ does it take the block to return to $x=0,$ and what is its speed at this point?

Key Concepts

Newton's Second Law

Newton's second law states that the net force acting on an object is equal to the mass of the object times its acceleration (F = ma). This principle is fundamental for analyzing the motion of objects under any force, including those that vary with time, by relating the applied forces to the resulting acceleration and subsequent motion.

Variable Forces and Time-Dependent Acceleration

When forces change over time, the acceleration of an object becomes a time-dependent function. In such cases, the instantaneous value of acceleration can be determined by dividing the force by the mass, and this function must be integrated or differentiated with respect to time to obtain velocity and position. This approach is essential for solving problems where the applied force is not constant.

Integration to Determine Velocity and Displacement

In problems with time-dependent acceleration, integration is used to derive the velocity function by integrating the acceleration function over time, and further integration of the velocity function yields the position function. These integrations must account for initial conditions, such as starting from rest, which help determine the constants of integration.

Identifying Extreme Points in Motion (Critical Points)

Finding the maximum displacement or turning point in an object’s motion typically involves identifying when the velocity becomes zero. At these critical points, the object momentarily stops before changing direction. Setting the velocity function to zero and solving for time yields the moment of maximum displacement, which is a crucial step in analyzing the complete motion of the object.

Transcript

00:02 In this problem, we are considering a block of mass, 2 kilograms initially at rest on a slippery horizontal surface, for it sure is no friction.

00:13 Starting at time equals zero, a horizontal force that's listed right here is applied to the block where we're given the values for those two, alpha and beta.

00:25 What is this largest velocity reached by the block? okay, so here is the equation we're going to use.

00:35 We're going to consider these equations and write our fx, our x component, and we'll get this equation.

00:44 Doing our math, you can follow through, take a look at this, and we're integrating through t and 0, vmax, and v at initial.

00:55 Okay, here we are.

01:00 I got a peek and see where i am here.

01:03 Okay.

01:06 So here we are.

01:07 Still solving, still solving, still solving, and here's our x at t, time t.

01:30 Okay, we're still working here, and then we've got putting our equations in here, and you can watch me just do these.

01:40 Feel free to take your time and go through these.

01:45 There i've subs to all the values, and here's my final answer for part a, 0 .593 meters.

01:56 For b, we're asked, well, no, i'm not done yet.

02:08 Okay, so here i had more of this question.

02:12 For a, our largest value of x, here was our magnitude of our force, and there's how long it took.

02:19 So i solved for all three of these here.

02:22 Let me get a nice bright number here.

02:27 Here's my distance.

02:42 Here's my four newtons.

02:45 Where's my 0 .133.

02:47 It's in here somewhere.

03:00 Let me take a look here...

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