- The velocity function is derived as \(v(t) = 3 - 4t\) by differentiating the position function \(s(t) = 3t - 2t^2\).
- Setting \(v(t) = 0\) reveals the particle is at rest at \(t = 0.75\).
- The particle moves in a positive direction for \(t \in [0, 0.75)\) where \(v(t) > 0\).
- The particle moves in a negative direction for \(t \in (0.75, \infty)\) where \(v(t) < 0\), with final intervals \(\boxed{[0, 0.75)}\) and \(\boxed{(0.75, \infty)}\).
Explanation
- Finding the Velocity Function
The position of the particle is given by the function \(s(t) = 3t - 2t^2\) for \(t \geq 0\). The velocity of a particle, denoted by \(v(t)\), is the instantaneous rate of change of its position with respect to time, which corresponds to the first derivative of the position function. By applying the power rule of differentiation to each term in the function, we find the velocity function as follows:
\[v(t) = \frac{d}{dt}(3t - 2t^2) = 3 - 4t\]
- Determining Critical Time Points
The particle changes its direction of motion at points where its velocity is zero. To find these critical time points, we set the velocity function equal to zero and solve for \(t\):
\[3 - 4t = 0 \implies 4t = 3 \implies t = \frac{3}{4} = 0.75\]
The particle is at rest at time \(t = 0.75\). This point divides the domain \(t \geq 0\) into two distinct intervals: \([0, 0.75)\) and \((0.75, \infty)\).
- Analyzing Direction of Motion
To determine the direction of motion, we evaluate the sign of the velocity function \(v(t) = 3 - 4t\) in each interval:
- For \(0 \leq t < 0.75\), we can pick a test point, such as \(t = 0\). Evaluating the velocity at this point gives \(v(0) = 3 - 4(0) = 3\), which is positive. Thus, the particle moves in a positive direction when \(0 \leq t < 0.75\).
- For \(t > 0.75\), we can pick a test point, such as \(t = 1\). Evaluating the velocity at this point gives \(v(1) = 3 - 4(1) = -1\), which is negative. Thus, the particle moves in a negative direction when \(t > 0.75\).
- Final Conclusion
Based on the velocity function and the sign analysis, we conclude the following:
- The velocity function is \(v(t) = 3 - 4t\).
- The particle moves in a positive direction on the interval \([0, 0.75)\).
- The particle moves in a negative direction on the interval \((0.75, \infty)\).
The final results for the intervals are \(\boxed{[0, 0.75)}\) for the positive direction and \(\boxed{(0.75, \infty)}\) for the negative direction.
Examples
In real-world applications, understanding the velocity of a moving object is essential for fields like robotics and logistics. For instance, if an automated delivery robot is moving along a path according to a specific position formula, knowing exactly when it transitions from moving forward (positive direction) to backward (negative direction) allows engineers to program precise navigation commands and avoid collisions, ensuring safe and efficient delivery of goods.