A fireworks shell is accelerated from rest to a velocity of 65.0 m/s over a distance of 0.250 m.
(a) How long did the acceleration last?
(b) Calculate the acceleration.
Solution:
We are given the following values:v_0=0\:\text{m/s}; v_f=65.0\:\text{m/s}; and \Delta x=0.250\:\text{m}.
We can immediately solve for the acceleration using the given values, so we are going to answer Part B first.
Part B
Solve for the acceleration first using the formula
\left(v_f\right)^2=\left(v_0\right)^2+2a\Delta x
We solve for acceleration in terms of the other variables.
a=\frac{\left(v_f\right)^2-\left(v_0\right)^2}{2\Delta x}
Substitute the given values
\begin{align*} a & =\frac{\left(v_f\right)^2-\left(v_0\right)^2}{2\Delta x} \\ a & = \frac{\left(65.0\:\text{m/s}\right)^2-\left(0\:\text{m/s}\right)^2}{2\left(0.250\:\text{m}\right)} \\ a & =8450\:\text{m/s}^2 \ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}
Part A
To solve for the time of this motion, we shall use the formula
v_f=v_0+at
Solving for time, t, in terms of the other variables we have.
t=\frac{v_f-v_0}{a}
We now substitute the values given, and the computed acceleration to find the time.
\begin{align*} t & =\frac{v_f-v_0}{a} \\ t & =\frac{65.0\:\text{m/s}-0\:\text{m/s}}{8450\:\text{m/s}^2} \\ t & =7.6922\:\times 10^{-3}\:\text{s}\ \qquad \ \color{DarkOrange} \left( \text{Answer} \right) \end{align*}
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