Friday, March 4, 2016

Kinematic diagram: Time vs velocity

Question:
The time vs velocity diagrams of a 10-kg object for $x$ and $y$ directions are given as shown. Describe each motion for each time interval.
(1) 0 < $t$ < 2 seconds
(2) 2 < $t$ < 3 seconds
(3) 3 < $t$ < 4 seconds
(4) at 2 seconds

Answer:
(1) From the diagram, the object moves at the constant velocity, 1 m/s, in $x$ direction, but it gets acceleration of $\frac{3}{2}$ m/s$^2$ in $y$ direction. The slope indicates the acceleration. Thus, it is exerted by 15 N in that direction because $F=ma$.

(2) For both directions, during this time interval, the object moves at the constant velocity. The diagrams show the flat slopes; namely, no acceleration and no force on the object.

(3) The accelerations from 3 s to 4 s are
\begin{equation*}
a_x = a_y = \frac{0-3}{4-3} = -3 \ \mathrm{m/s^2}
\end{equation*}
The magnitude of the negative acceleration in both directions is equal, so the force is directed in opposite to 45 degrees on $x-y$ plane. The magnitude of force is given by
\[
|F| = 10 \times \sqrt{3^2+3^2} = 42 \ \mathrm{N}
\]

(4) At 2 seconds, the velocity in $x$ direction is suddenly changed from 1 m/s to 3 m/s. This indicates the change in momenta; namely, there is an impulse in positive $x$ direction. The impulse is calculated as
\[
I = mv_f - mv_i = 10 \cdot 3 - 10 \cdot 1 = 20 \ \mathrm{N\cdot s}
\]

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