Question

The equation of motion of a particle is \(s = {t^4} - 2{t^3} + {t^2} - t\), where \(s\) is in meters and \(t\) is in seconds. Find

(a) the velocity and acceleration as functions of \(t\),

(b) the acceleration after 1 s.

(c)Graph the position, velocity, and acceleration functions on the same screen.

Short answer

1. The velocity is \(\upsilon \left( t \right) = 4{t^3} - 6{t^2} + 2t - 1\)and the acceleration is\(a\left( t \right) = 12{t^2} - 12t + 2\).
2. The acceleration after 1 sec is 2m/s².
3. Visualization of the graph of the functions is shown below:

Step-by-step solution

(a)Step 1: Find first derivative and second derivative

The given function of motion is \(s = {t^4} - 2t + {t^2} - t\). On differentiating this function with respect to \(t\), the following expression for velocity is obtained:

\(\begin{aligned}\upsilon \left( t \right) &= s'\left( t \right)\\ &= \frac{d}{{dt}}\left( {{t^4} - 2t + {t^2} - t} \right)\\ &= \frac{d}{{dt}}\left( {{t^4}} \right) - \frac{d}{{dt}}\left( {2t} \right) + \frac{d}{{dt}}\left( {{t^2}} \right) - \frac{d}{{dt}}\left( t \right)\\ &= 4{t^3} - 6{t^2} + 2t - 1\end{aligned}\)

Thus, the velocity is \(\upsilon \left( t \right) = 4{t^3} - 6{t^2} + 2t - 1\).

The first derivative of the function is \(s' = 4{t^3} - 6{t^2} + 2t - 1\). On differentiating it again, with respect to \(t\) axis, we obtain the acceleration, as follows:

\(\begin{aligned}s'' &= a\left( t \right)\\ &= \frac{d}{{dt}}\left( {4{t^3} - 6{t^2} + 2t - 1} \right)\\ &= 12{t^2} - 12t + 2\end{aligned}\)

Thus, the acceleration is \(a\left( t \right) = 12{t^2} - 12t + 2\).

(b)Step 2: Find acceleration after 1 sec

Plug \(t = 1\)into \(a\left( t \right)\) to find the acceleration after \(1\) second, as shown below:

\(\begin{aligned}a\left( 1 \right) &= 12{\left( 1 \right)^2} - 12\left( 1 \right) + 2\\ &= 2\end{aligned}\)

Thus, the acceleration after 1 sec is 2m/s².

(c)Step 3: Draw the graphs

The procedure to draw the graph of the above equation by using the graphing calculator is as follows:

Draw the graph of the functions\(\upsilon \left( t \right) = 4{t^3} - 6{t^2} + 2t - 1\),\(a\left( t \right) = 12{t^2} - 12t + 2\) and \(s = {t^4} - 2t + {t^2} - t\)by using the graphing calculator as shown below:

1. Open the graphing calculator. Select the “STAT PLOT” and enter the equation\(4{X^3} - 6{X^2} + 2X - 1\)in the\({Y_1}\)tab,\(12{X^2} - 12X + 2\)in \({Y_2}\)tab.
2. Enter the “GRAPH” button in the graphing calculator.

Visualization of the graph of the functions is shown below: