Analyzing the Motion of a Particle: Distance, Velocity, and Acceleration at t 2 Seconds

The motion of a particle is described by the position equation s 2t^4 - frac{t^3}{6} - 2t^2, where s represents the position in feet and t represents the time in seconds. This article dives into how to calculate the distance, velocity, and acceleration of the particle at t 2 seconds, following a step-by-step method. By the end, you will have a comprehensive understanding of the motion characteristics of the particle.

Step 1: Calculating Distance

To find the distance traveled by the particle at t 2 seconds, substitute t 2 into the position equation.

The equation is:

[pounds]s 2t^4 - frac{t^3}{6} - 2t^2[/pounds]

Substituting t 2 into the equation:

begin{align*} s(2) 2(2^4) - frac{(2^3)}{6} - 2(2^2) 2 times 16 - frac{8}{6} - 2 times 4 32 - frac{4}{3} - 8 32 - frac{4}{3} - 8 frac{96}{3} - frac{4}{3} - frac{24}{3} frac{96 - 4 - 24}{3} frac{72}{3} 24 text{ feet} end{align*}

Therefore, the distance traveled by the particle at t 2 seconds is 24 feet.

Step 2: Calculating Velocity

The velocity of the particle is the first derivative of the position function with respect to time t. The derivative of the position function is:

begin{align*} v(t) frac{ds}{dt} frac{d}{dt}left(2t^4 - frac{t^3}{6} - 2t^2right) 8t^3 - frac{3t^2}{6} - 4t 8t^3 - frac{t^2}{2} - 4t end{align*}

Substituting t 2 into the velocity equation:

begin{align*} v(2) 8(2^3) - frac{(2^2)}{2} - 4(2) 8(8) - frac{4}{2} - 8 64 - 2 - 8 54 text{ feet/second} end{align*}

Hence, the velocity of the particle at t 2 seconds is 54 feet/second.

Step 3: Calculating Acceleration

The acceleration of the particle is the second derivative of the position function with respect to time t. The second derivative of the position function is:

begin{align*} a(t) frac{dv}{dt} frac{d^2}{dt^2}left(2t^4 - frac{t^3}{6} - 2t^2right) 24t^2 - frac{3t^2}{6} - 4 24t^2 - frac{t^2}{2} - 4 end{align*}

Substituting t 2 into the acceleration equation:

begin{align*} a(2) 24(2^2) - frac{(2^2)}{2} - 4 24(4) - frac{4}{2} - 4 96 - 2 - 4 90 text{ feet/second}^2 end{align*}

The acceleration of the particle at t 2 seconds is 90 feet/second2.

Conclusion

Summarizing the results:

The distance traveled at t 2 seconds is 24 feet. The velocity at t 2 seconds is 54 feet/second. The acceleration at t 2 seconds is 90 feet/second2.

By breaking down the steps of differentiation and substituting the specific time, we can accurately calculate and understand the motion of the particle at any given moment.