Question

Differential settlement of a segment of roadway causes a small drop of magnitude \(h\). Repairs are performed effecting a transition ramp of length \(L\) between the two segments of the roadway. A vehicle travels at constant speed \(v_{0}\) along a flat road when it encounters the drop. Assuming that the driver maintains the same horizontal speed throughout his motion, determine the vertical motion of the vehicle during and after it encounters the drop. The vehicle never leaves the roadway and the vertical speed due to gravity is negligible.

Short answer

Answer: The vehicle's vertical motion when it encounters the ramp is given by the vertical component of velocity, which is calculated as \(v_{y} = v_{0} \sin(\theta)\), where \(\theta\) is the angle of the ramp and \(v_{0}\) is the constant horizontal speed.

Step-by-step solution

Determine the angle of the ramp

To determine the angle of the ramp, we can use basic trigonometry by knowing the length \(L\) and the height \(h\) of the ramp. We can use the inverse tangent function, called arctangent, to find the angle \(\theta\): $$ \theta = \arctan\left(\frac{h}{L}\right) $$

Calculate the vertical component of velocity

Now that we have the angle of the ramp, we can calculate the vertical component of velocity \(v_{y}\) using trigonometry, since the vehicle's horizontal speed \(v_{0}\) remains constant. We can find \(v_{y}\) by: $$ v_{y} = v_{0} \sin(\theta) $$ Step 2: Analyze the vertical motion on the flat road, ramp and after encountering the drop

Flat road and ramp motion

On the flat road and while on the ramp, the vehicle's vertical motion is constant due to the lack of vertical forces acting on it. Therefore, the vertical velocity is 0 on the flat road, and it is \(v_{y}\) calculated in Step 1 when on the ramp.

After encountering the drop

After the vehicle has encountered the drop and is back on a level road, its vertical velocity would be 0, as there is no vertical acceleration due to gravity. The vehicle has transitioned safely over the drop and maintains its horizontal speed \(v_{0}\). In conclusion, during the vehicle's travel on the flat road and after encountering the drop, it does not experience any vertical motion. However, while on the ramp, the vehicle's vertical velocity is \(v_{y} = v_{0} \sin(\theta)\). The vehicle maintains its constant horizontal speed \(v_{0}\) throughout its entire motion.

Key concepts

Differential Settlement

Differential settlement refers to the scenario where different parts of a structure or a roadway settle by varying amounts. This commonly occurs due to uneven loading, variances in soil properties, or moisture content changes underground. In our context, a segment of roadway experiencing differential settlement results in a drop of magnitude h. This drop requires a transition in the form of a ramp to facilitate smooth vehicle travel and prevent immediate impact or stress on the vehicle. Understanding how differential settlement impacts vehicle motion is crucial for civil engineers when designing roadways, particularly to ensure safety and comfort during travel.

Differential settlement necessitates the calculation of an appropriate transition ramp angle to accommodate the physical changes without disrupting the vehicle's motion more than necessary. In our exercise, the length of the ramp and the height of the drop give us the necessary parameters to calculate the angle of inclination using trigonometry.

Trigonometry in Engineering

Trigonometry plays a fundamental role in engineering, aiding in the analysis and design of various structures, including roads, bridges, and buildings. In our vehicle vertical motion analysis, trigonometry helps determine the angle of a ramp, which is a real-life engineering challenge. The angle of the ramp, derived using the relationship between the opposite side (the drop) and the adjacent side (the ramp length), directly influences the design and safety of the roadway. Engineers utilize trigonometric functions, such as sine and cosine, to resolve forces and calculate velocities, ensuring stable structural designs and smooth mechanical operations.

Through the application of trigonometric principles, engineers can predict and mitigate potential issues in structures and vehicles, safeguarding against accidents or structural failure. By examining real-world problems like differential settlement, students gain insight into the vital role trigonometry plays in engineering solutions.

Arctangent Function

The arctangent function, represented as arctan or tan^-1, is one of the inverse trigonometric functions vital in solving for angles when only side lengths of a right triangle are known. As seen in the exercise, the arctangent of the ratio h/L calculates the ramp angle \(\theta\).

Being familiar with the arctangent function is essential for students since it is heavily utilized in both academic and professional engineering to find unknown angles. Understanding its utility helps students tackle a range of problems from simple angles to assessing the rate of change, slopes, or addressing more complex system dynamics where angles vary over time.

Vertical Component of Velocity

The vertical component of velocity, v_y, is a part of the overall velocity vector of an object moving in two or three dimensions. It represents the rate at which the object is moving along the vertical axis. In our vehicle motion analysis, while the vehicle maintains a constant horizontal speed, its vertical speed changes as it travels up and down the ramp.

To calculate this vertical component, we use trigonometry, specifically the sine function of the angle of the incline or decline. This component is crucial for understanding the performance of vehicles on inclined surfaces and for ensuring that the designs can accommodate the changes in vertical forces that the vehicles will experience. The concept ties into real-world engineering applications such as designing road gradients and determining how they affect vehicle performance and safety.

Constant Speed Motion

Constant speed motion refers to the motion where an object travels at a uniform speed along a straight path. In the context of the exercise, the vehicle maintains a constant horizontal speed v₀, uninterrupted by the vehicle's vertical movement due to the inclined ramp. This implies that no acceleration occurs along the horizontal path of travel.

This concept is essential, as it simplifies the analysis of motion, allowing the use of basic trigonometry to determine other aspects of the vehicle's motion such as vertical components of velocity. In practical engineering, constant speed motion is idealized as many real-world factors can introduce variations in speed. Nevertheless, it provides a foundational understanding of motion, which can then be expanded with more complex physics to cater to real-world variables.