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Solved Problem on Kinematics

A motorcyclist is moving in the opposite direction of a reference frame, the magnitude of its initial speed is 25 m/s, at the initial time its position is -150 m, and the magnitude of deceleration is 2 m/s

a) The equation of displacement as a function of time;

b) The equation of velocity as a function of time;

c) The instant in which it passes through the origin of the reference frame;

d) The instant that its speed is zero.

Problem data

- the initial speed of the rider: |
*v*_{0}| = 25 m/s; - acceleration of the motorcyclist: |
*a*| = 2 m/s^{2}; - position in the initial instant:
*S*_{0}= −150 m.

We chose a reference frame oriented to the right.

Solution

a) The equation of displacement as a function of time is of the type

\[ \bbox[#99CCFF,10px]
{S=S_{0}+v_{0}t+\frac{a}{2}\;t^{2}}
\]

The initial position is already given in the problem,
\[
S=-150-25t+\frac{2}{2}\;t^{2}
\]

\[ \bbox[#FFCCCC,10px]
{S=-150-25t+t^{2}}
\]

b) For the equation of velocity as a function of time is of type

\[ \bbox[#99CCFF,10px]
{v=v_{0}+at}
\]

with the problem data, we have
\[ \bbox[#FFCCCC,10px]
{v=-25t+2t}
\]

c) When the motorcyclist passes through the origin we have

\[
0=-150-25\;t+t^{2}
\]

This is a
the solution of the *Quadratic Equation*
\(
t^{2}-25\;t-150=0
\)

\[
\Delta =b^{2}-4ac=(\;-25\;)^{2}-4 \times 1 \times (\;-150\;)=625+600=1225\\
{}\\
t=\frac{-b\pm \sqrt{\;\Delta \;}}{2a}=\frac{-(\;-25\;)\pm \sqrt{\;1225\;}}{2 \times 1}=\frac{25\pm 35}{2}
\]

the two roots of the equation will be

\( t_{1}=30\;\text{s} \)
and
\( t_{2}=-5\;\text{s} \)

Since there is no negative time, we neglected the second root, it will go through the origin at

d) When the speed of the motorcycle cancels we have

\[
0=-25+2t
\]

This is a
\[
2t=25\\
t=\frac{25}{2}
\]

\[ \bbox[#FFCCCC,10px]
{t=12.5\;\text{s}}
\]

This is the instant that the motorcyclist changes direction and begins to move in the same direction of the referential
frame until passing through the origin.
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Fisicaexe - Physics Solved Problems by Elcio Brandani Mondadori is licensed under a Creative Commons Attribution-NonCommercial-ShareAlike 4.0 International License .