Problem 45 - Thomas Calculus

Question

The equations in Exercises 45 and 46 give the position \(s=f(t)\) of a body moving on a coordinate line \((s \text { in meters, } t \text { in seconds). Find the }\) body's velocity, speed, acceleration, and jerk at time \(t=\pi / 4\) sec. $$ s=2-2 \sin t $$

Step-by-Step Solution

Answer

Velocity: \(-\sqrt{2}\) m/s; Speed: \(\sqrt{2}\) m/s; Acceleration: \(\sqrt{2}\) m/s²; Jerk: \(\sqrt{2}\) m/s³.

Step 1: Find the velocity

To find the velocity of the body, we need to take the first derivative of the position function with respect to time. Given the position function \(s = 2 - 2 \sin t\), we have:\[v = \frac{ds}{dt} = \frac{d}{dt}(2 - 2 \sin t) = -2 \cos t\]Now, substitute \(t = \pi/4\) into the velocity equation:\[v(\pi/4) = -2 \cos(\pi/4) = -2 \left(\frac{\sqrt{2}}{2}\right) = -\sqrt{2}\]Thus, the velocity at \(t = \pi/4\) seconds is \(-\sqrt{2}\) m/s.

Step 2: Find the speed

Speed is the absolute value of velocity. Thus, from the velocity found in Step 1:\[\text{Speed} = |-\sqrt{2}| = \sqrt{2} \]So, the speed at \(t=\pi/4\) seconds is \(\sqrt{2}\) m/s.

Step 3: Find the acceleration

To find the acceleration, take the derivative of the velocity function. From Step 1, we have the velocity function \(v = -2 \cos t\):\[a = \frac{dv}{dt} = \frac{d}{dt}(-2 \cos t) = 2 \sin t\]Substitute \(t = \pi/4\) into the acceleration equation:\[a(\pi/4) = 2 \sin(\pi/4) = 2 \left(\frac{\sqrt{2}}{2}\right) = \sqrt{2}\]Thus, the acceleration at \(t = \pi/4\) seconds is \(\sqrt{2}\) m/s².

Step 4: Find the jerk

Jerk is the derivative of acceleration. From Step 3, we know that the acceleration is \(a = 2 \sin t\):\[j = \frac{da}{dt} = \frac{d}{dt}(2 \sin t) = 2 \cos t\]Substitute \(t = \pi/4\) into the jerk equation:\[j(\pi/4) = 2 \cos(\pi/4) = 2 \left(\frac{\sqrt{2}}{2}\right) = \sqrt{2}\]Thus, the jerk at \(t = \pi/4\) seconds is \(\sqrt{2}\) m/s³.

Key Concepts

Understanding Derivatives in Motion

Derivatives play a crucial role in calculus, especially when analyzing motion. When a function describes the position of a moving object, as in \(s = 2 - 2 \sin t\), taking its derivative allows us to find the velocity. Velocity is the rate of change of position with respect to time. For instance, by differentiating \(s\) with respect to \(t\), we are observing how quickly the position \(s\) changes as \(t\) varies. This is similar to finding the slope of a tangent line at any given point on the position-time graph.

In our example, the derivative of \(2 - 2 \sin t\) with respect to \(t\) results in \(-2 \cos t\). Here, \(-2 \cos t\) signifies that the velocity depends on the cosine function, showcasing predictable oscillatory motion. Calculating derivatives is essential whenever you want to find related rates, such as acceleration or jerk, which are subsequent derivatives of the velocity and acceleration functions respectively.

Calculating Velocity

Velocity is the first derivative of a position function in terms of time. It represents the speed of an object moving in a direction. In our problem, we find the velocity by taking the derivative of the position function \(s = 2 - 2 \sin t\). The resulting velocity function is \(v = -2 \cos t\). This function tells us how fast the object is moving and in which direction at a specific time.

When we substitute \(t = \pi/4\) into the velocity function, we calculate \(-2 \cos(\pi/4) = -\sqrt{2}\), meaning that at \(t = \pi/4\) seconds, the velocity is \(-\sqrt{2}\) m/s. The negative sign indicates direction, showing the object moves in the negative direction along the coordinate line.

Kinematics: The Study of Motion

Kinematics is the branch of physics that describes the motion of objects without considering the forces that cause them. It involves variables such as position, velocity, acceleration, and jerk. These are crucial for understanding how an object moves along a path over time.

In our given function \(s = 2 - 2 \sin t\), the position of a body is defined in terms of time \(t\). By deriving this function multiple times, we sequentially determine velocity, acceleration, and jerk. Acceleration, the second derivative, details how the velocity changes, found as \(a = 2 \sin t\). Jerk, the third derivative \(j = 2 \cos t\), indicates how acceleration changes over time. Each step further adds depth to modeling and understanding the body's motion.

Exploring Trigonometric Functions

Trigonometric functions, like sine and cosine, are essential in describing periodic phenomena, including waveforms or oscillatory motion. In our exercise, both the position and derived functions (velocity, acceleration, jerk) involve trigonometric functions. This is because the original equation \(s = 2 - 2 \sin t\) is periodic, showing that the position of the object oscillates over time.

In our problem, the velocity is expressed as \(-2 \cos t\), an oscillatory function, resulting from differentiating the sine function. Similarly, acceleration is \(2 \sin t\), and jerk \(2 \cos t\); all rely on trigonometric identities for their calculation. Understanding these functions helps predict and analyze the motion of objects in continuous, cyclic scenarios.