Motion Equation: Initial Velocity & Max Displacement

Hey, physics enthusiasts and curious minds! Today, we're diving deep into the fascinating world of motion. We're going to break down a problem involving a particle cruising along a straight line. The equation governing its movement is given by S = 18t + 3t² - 2t³, where 'S' represents the position in meters and 't' is the time in seconds. Our mission, should we choose to accept it, is to figure out a couple of key things: (a) the particle's velocity right at the start, and (b) the exact moment when the particle hits its maximum displacement.

Initial Velocity: Let's Get Started!

So, the first part of our challenge is to determine the initial velocity of the particle. Remember, velocity is the rate of change of displacement with respect to time. In calculus terms, it's the first derivative of the displacement equation. Thus, we need to find the derivative of S with respect to t. Given the displacement equation $S = 18t + 3t^2 - 2t^3$, we can find the velocity v by differentiating S with respect to t: $v = dS/dt = 18 + 6t - 6t^2$. This equation describes the velocity of the particle at any time t. Now, to find the initial velocity, we simply plug in t = 0 into the velocity equation, because we're interested in the velocity at the very beginning of the motion. Therefore, $v(0) = 18 + 6(0) - 6(0)^2 = 18 m/s$. So, the particle's initial velocity is a solid 18 meters per second. This means that at the very start, the particle is already moving at a speed of 18 m/s in the positive direction. This is a crucial piece of information as it sets the stage for the rest of the particle's motion. Understanding the initial conditions is always the first step in analyzing any dynamic system. From here, we can explore how the velocity changes over time due to the acceleration (or deceleration) described by the subsequent terms in the velocity equation. As time progresses, the terms involving t and t² will start to influence the velocity, causing it to either increase or decrease depending on their signs and magnitudes. But for now, we've successfully pinpointed the starting speed of our particle, giving us a solid foundation for the rest of our analysis. Cool, right?

Time to Maximum Displacement: Finding the Peak

Next up, let's pinpoint the time when our particle reaches its maximum displacement. Maximum displacement occurs when the particle momentarily stops before changing direction. In other words, at the point of maximum displacement, the velocity of the particle is zero. We already found the equation for the velocity in the previous section: $v = 18 + 6t - 6t^2$. To find the time at which the velocity is zero, we need to solve the quadratic equation $18 + 6t - 6t^2 = 0$. First, let's simplify the equation by dividing all terms by 6: $3 + t - t^2 = 0$. Rearranging the terms, we get $t^2 - t - 3 = 0$. Now we can use the quadratic formula to solve for t: $t = [-b ± sqrt(b^2 - 4ac)] / (2a)$. In our equation, a = 1, b = -1, and c = -3. Plugging these values into the quadratic formula, we get $t = [1 ± sqrt((-1)^2 - 4(1)(-3))] / (2(1)) = [1 ± sqrt(1 + 12)] / 2 = [1 ± sqrt(13)] / 2$. So, we have two possible values for t: $t = (1 + sqrt(13)) / 2$ and $t = (1 - sqrt(13)) / 2$. Since time cannot be negative, we discard the second solution, which would give us a negative value for t. Therefore, the time at which the particle reaches its maximum displacement is $t = (1 + sqrt(13)) / 2$ seconds. Approximating the square root of 13 as roughly 3.6, we get $t ≈ (1 + 3.6) / 2 = 4.6 / 2 = 2.3$ seconds. This result indicates that the particle reaches its furthest point from the origin at approximately 2.3 seconds. Understanding this point is critical because it tells us when the particle transitions from moving away from its starting point to returning towards it. The time of maximum displacement helps us visualize the particle's journey and understand its overall motion pattern. Moreover, it’s a great example of how mathematical tools like calculus and the quadratic formula can be applied to solve real-world physics problems. Keep practicing and exploring – the world of physics is full of such fascinating insights!

Wrapping It Up

Alright, guys, let's recap what we've discovered. We started with the equation of motion for a particle moving in a straight line: S = 18t + 3t² - 2t³. We then successfully determined that:

• The initial velocity of the particle is 18 m/s. This is the speed at which the particle starts moving at time t=0.
• The time at which the particle reaches its maximum displacement is approximately 2.3 seconds. This is when the particle momentarily stops before reversing direction.

By using calculus to find the first derivative of the displacement equation (which gave us the velocity equation) and then solving for when the velocity is zero, we were able to answer both parts of the problem. Remember, the initial velocity is found by evaluating the velocity equation at t=0, and the time of maximum displacement is found by setting the velocity equation to zero and solving for t. These are fundamental concepts in kinematics, and mastering them will help you tackle a wide range of motion-related problems. Keep practicing, and don't be afraid to dive deeper into the concepts. Physics is an adventure, and every problem is a new opportunity to learn something awesome! Stay curious and keep exploring, Plastik Magazine readers!