Question

FIGURE EX2.32 shows the acceleration graph for a particle that starts from rest at t = 0 s. What is the particle’s velocity at t = 6 s?

Short answer

The velocity of a particle at $t=6s$is 30 m s^-1.

Step-by-step solution

Given information

The initial velocity of a particle, $v_0=0\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$

Initial time,

The velocity of a particle.

The velocity of a particle between time zero and 6 s is the area under the curve from time zero to $t_f=6s$. In this case, the curve is a triangle.

Thus, the area of a triangle is calculated as follows.

$$\begin{gathered} ∆v=\frac{1}{2}\times 6s\times 10\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s^2$}\right. \\ =30\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right. \end{gathered}$$

Now,

The velocity of a particle at $t_f=6s$is given by

area under the acceleration curve between and

$$\begin{gathered} v_f=0\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.+30\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right. \\ =30\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right. \end{gathered}$$

Therefore, the velocity of a particle at $t=6s$is $30\raisebox{1ex}{$m$}\!\left/ \!\raisebox{-1ex}{$s$}\right.$.