Question

A $\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{kg}$particle moves along an xaxis, being propelled by a variable force directed along that axis. Its position is given by role="math" localid="1657014461331" $\mathit{x}\mathbf{=}\mathbf{3}\mathbf{.}\mathbf{0}\mathbf{m}\mathbf{+}\mathbf{(}\mathbf{4}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{m}\mathbf{/}\mathbf{s}\mathbf{)}\mathit{t}\mathbf{+}\mathit{c}{\mathit{t}}^{\mathbf{2}}\mathbf{-}\mathbf{(}\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{}\mathbf{m}\mathbf{/}\mathit{s}\mathbf{)}{\mathit{t}}^{\mathbf{3}}$, with xin meters and tin seconds. The factor cis a constant. Atthe force on the particle has a magnitude of $\mathbf{36}\mathbf{N}$and is in the negative direction of the axis. What is c?

Short answer

The value of constant c is$+9.0$

Step-by-step solution

The given data

1. Mass of particle, $m=2.0\mathrm{kg}$
2. Position of a particle in expression, $x=3.0+4t+c{t}^2-2t^3$
3. Magnitude of force at t= 3.0 s, $F=-36\mathrm{N}$which is in a negative direction

Understanding the concept of motion

The acceleration of the object is equal to the time rate of change of velocity. The acceleration can be found by differentiating the velocity vector with respect to time or differentiating the displacement vector twice with respect to time.

As we are given the function for the position of a particle if we differentiate that for time, we will get twice the acceleration. As we have got the value of force at a particular time, using Newton’s 2nd law, we can find the value for c.

Formula:

The force on a body due to Newton’s second law,$\overrightarrow{F}=m\overrightarrow{a}$ (i)

Here, $\overrightarrow{F}$is net force, is mass of the object, and$\overrightarrow{a}$is the acceleration.

The acceleration of a body in motion,$a=\frac{d^{2}x}{d{t}^2}$ (ii)

Calculation of the value of constant c

Taking the derivative of the given position equation$x=3.0+4t+c{t}^2-2t^3$with respect to time, we get the velocity as follows:

role="math" localid="1657015128101" $v=0+4+2ct-6t^2..............\left(a\right)$