Q82P - Fundamentals Of Physics

Question

Figure 2-44 gives the acceleration a versus time t for a particle moving along an x axis. The ‘a’ axis scale is set by ${\mathit{a}}_{\mathbf{s}}\mathbf{=}\mathbf{12}\mathbf{.}\mathbf{0}\mathbf{ }\mathbf{m}\mathbf{/}{\mathbf{s}}^{\mathbf{2}}$. At $\mathit{t}\mathbf{=}\mathbf{‐}\mathbf{2}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{s}$, the particle’s velocity is 7.0 m/s . What is its velocity at $\mathit{t}\mathbf{=}\mathbf{6}\mathbf{.}\mathbf{0}\mathbf{ }\mathit{s}$?

Step-by-Step Solution

Answer

The velocity of the particle at $t=6.0 s$ is  $39 \mathrm{m}/\mathrm{s}$

Step 1: Given information

$$\begin{gathered} a_s=12.0 \mathrm{m}/{\mathrm{s}}^2 \\ t=-2.0 \mathrm{s} \\ {v}_0=7.0 \mathrm{m}/\mathrm{s} \end{gathered}$$

Step 2: Kinematic Equations of Motion

Kinematics is the study of how a system of bodies moves without taking into accounts the forces or potential fields that influence the motion. The equations which are used in the study are known as kinematic equations of motion.

 

Formula:

 

The final velocity is given by

$v_f=v_0+{\int }_{-2}^{6}a dt$

Step 3: Calculations for the velocity of the particle

The velocity of the particle at $t=6.0 sec$ can be calculated as,

$v_f=v_0+{\int }_{-2}^{6}a dt$

From the given graph the acceleration as a function of t can be written as,

$a=2t$

So, 

$$\begin{gathered} v_f=v_0+{\int }_{-2}^{6}2t dt \\ {v}_f=7+2{\left[\frac{t^2}{2}\right]}_{-2}^6 \\ {v}_f=7+32 \\ =39 \mathrm{m}/\mathrm{s} \end{gathered}$$

 

Hence, the velocity of the particle will be 39 m/s