91Ó°ÊÓ

When examining fluid dynamics, it is important to understand how acceleration behaves along a streamline, which is the path followed by a fluid particle. The acceleration along the streamline, also known as tangential acceleration, reflects the change in speed of the fluid along its path.

In our problem, the velocity of water in the channel is initially described by the function \( V = 4(1 + 0.1t) \) ft/s for \( 0 \leq t \leq 20 \). Acceleration along the streamline can be identified by taking the derivative of this velocity with respect to time \( t \). This derivative tells us how the speed changes over time. For example, at \( t = 10 \) seconds, by differentiating the velocity equation, we find an acceleration of \( a_s = 0.4 \text{ ft/s}^2 \).

For times \( t \geq 20 \) seconds, the velocity becomes constant at \( V = 12 \text{ ft/s} \). Here, the acceleration along the streamline reduces to zero since the speed no longer changes. This aspect highlights the important concept that acceleration is linked to changing velocities.

Normal acceleration, or radial acceleration, occurs perpendicular to the flow direction along a curved path. It indicates how quickly the direction of the fluid particle's velocity vector is changing, which is crucial for understanding the behavior in curved channels.

The formula for calculating normal acceleration is \( a_n = \frac{V^2}{R} \), where \( V \) is the velocity and \( R \) is the radius of curvature of the path. For instance, at \( t = 10 \text{ seconds} \) with a computed velocity of \( V = 8 \text{ ft/s} \) and a given radius \( R \) of 50 ft, normal acceleration comes out to be \( 1.28 \text{ ft/s}^2 \).

When \( t = 30 \) seconds, the velocity is 12 ft/s, and using the same formula, we derive \( a_n = 2.88 \text{ ft/s}^2 \). This acceleration remains even when the speed is constant, illustrating how curvature affects the motion of the fluid.

Net acceleration is a comprehensive measure that combines both the tangential and normal components of acceleration. It provides the total rate of change of velocity of the fluid particle, including both changes in speed and changes in direction.

Net acceleration is calculated using the Pythagorean theorem: \( a = \sqrt{a_s^2 + a_n^2} \). At \( t = 10 \) seconds, using our calculated values, the net acceleration is approximately \( 1.34 \text{ ft/s}^2 \). This value is the magnitude of the vector sum of both acceleration components.

For \( t = 30 \) seconds, since the velocity is constant and thus \( a_s = 0 \), the net acceleration equals the normal component, \( 2.88 \text{ ft/s}^2 \). Understanding net acceleration is key to predicting the overall motion and behavior of the fluid in the channel.