Question

Two forces are exerted on an object of mass in thex direction as illustrated in the free-body diagram. (Intro 1 figure) Assume that these are the only forces acting on the object.

Figure 1

1.) Which of the curves labeled A to D on the graph (Part A figure) could be a plot of $v_x(t)$ , the velocity of the object in the x direction as a function of time?

A

B

C

or

D

Figure A

2.) Which of the curves labeled A to D on the graph (Part B figure) could be a plot of , the position of the object along the x axis as a function of time?

A

B

C

or

D

Figure B

Answer 1

Concepts and reason

The concepts used to know the variation of the position and velocity with time are Newton’s second law of motion and the kinematic equations.

The expression for the velocity of the object can be written by using the Newton’s second law of motion and the kinematic equations.

The expression for the position of the object can be determined by using the relation between the position and the velocity of the object.

Fundamentals

Newton’s second law of motion states that the acceleration of the object is directly proportional to the net force acting on the object and inversely proportional to the mass of the object.

The kinematic equation which relates final velocity v, initial velocity ${v_0},$acceleration a, and position x after time t is given as follows:

$\begin{array}{l}\\v = {v_0} + at\\\\x = {v_0}t + \frac{1}{2}a{t^2}\\\end{array}$

(1)

From the free body diagram, the net force acting on the object decreases towards right and increases towards left.

According to the Newton’s second law of motion, the acceleration of the object decreases with decrease in the net force acting on the object.

The expression for the velocity of the object at time t is given by,

${v_x}\left( t \right) = {v_0} - at$

Here, ${v_0}$ is the initial velocity of the object and a is the acceleration of the object. The negative sign represents the decrease in the acceleration of the object.

Rearrange the above equation as follows:

${v_x}\left( t \right) = - at + {v_0}$

The equation of the line is given by,

$y = mx + c$

Here, m is the slope, c is the y- intercept, y is the y coordinate, and x is the x coordinate.

Compare equation ${v_x}\left( t \right) = {v_0} - at$ with line equation $y = mx + c$ to find the slope m of line.

$m = - a$

Hence, the equation ${v_x}\left( t \right) = {v_0} - at$ is a linear curve with decreasing slope.

In figure A, the curve A represents the linear curve with zero slope, curve C represents the non- linear curve which is a parabola, and curve D represents the linear curve with increasing slope, and the curve B is a linear curve with decreasing slope.

(2)

The expression for the velocity of the object in x- direction as a function of time t is given by,

${v_x}\left( t \right) = {v_0} - at$

The relation between the velocity and the position of the object is given by,

${v_x}\left( t \right) = \frac{{dx\left( t \right)}}{{dt}}$

Here, $x\left( t \right)$ is the position of the object.

Substitute $\frac{{dx\left( t \right)}}{{dt}}$ for ${v_x}\left( t \right)$ in equation ${v_x}\left( t \right) = {v_0} - at$ as follows:

$\frac{{dx\left( t \right)}}{{dt}} = {v_0} - at$

Integrate the above equation as follows:

$\begin{array}{c}\\\int {\frac{{dx\left( t \right)}}{{dt}}dt} = \int {\left( {{v_0} - at} \right)dt} \\\\x\left( t \right) = {v_0}t - a\left( {\frac{{{t^2}}}{2}} \right)\\\end{array}$

Rearrange the above equation as follows:

$x\left( t \right) = - a\left( {\frac{{{t^2}}}{2}} \right) + {v_0}t$

The equation for the parabola is given by,

$y = a{x^2} + bx + c$

Here, a, b, and c are the coefficients.

The orientation of the parabola depends on the value of a.

Compare equation $x\left( t \right) = - a\left( {\frac{{{t^2}}}{2}} \right) + {v_0}t$ with equation $y = a{x^2} + bx + c$ as follows:

$\begin{array}{c}\\a = - \frac{a}{2}\left( {a < 0} \right)\\\\b = {v_0}\\\\c = 0\\\end{array}$

Hence, the equation $x\left( t \right) = - a\left( {\frac{{{t^2}}}{2}} \right) + {v_0}t$ is a parabola with opening downward.

In figure B, the curve A represents the linear curve with negative slope, curve B represents the linear curve with zero slope, and curve C represents the parabola with opening upward.

The curve D is a parabola with opening downward.

Ans: Part 1

The plot of velocity of the object as a function of time is B.

Answer 2

answer to 1 is B 

answer to 2 is idk because the guy who answered with a bunch of equations is stupid and ik ur here cause of masteringphysics and u wanna look up answers, sadly you are going to have to guess answer 2 because people like the guy who answered are so extra to the point where they didn't even include an answer

Answer 3

ANSWERS:

1: is A

2: is D (Since velocity is a negative slope, the interval would be a parabola, hence the answer being D)

Answer 4

1. is B
   

2. is D