Problem 10 - Basic Technical Mathematics with Calculus

Question

In designing a highway, a civil engineer must determine the length of a highway on-ramp for cars going onto the ramp at \(25 \mathrm{km} / \mathrm{h}\) and entering the highway at \(95 \mathrm{km} / \mathrm{h}\) in \(12.0 \mathrm{s}\). What minimum length should the on-ramp be?

Step-by-Step Solution

Answer

The minimum on-ramp length should be approximately 200 meters.

Step 1: Convert Speeds to Consistent Units

First, we need to convert the speeds from km/h to m/s, as we need consistent units for calculation. \(25 \text{ km/h} = \frac{25 \times 1000}{3600} \text{ m/s} = 6.94 \text{ m/s}\) and \(95 \text{ km/h} = \frac{95 \times 1000}{3600} \text{ m/s} = 26.39 \text{ m/s}\).

Step 2: Determine Acceleration

Using the formula for constant acceleration \(a = \frac{v_f - v_i}{t}\), where \(v_f = 26.39 \text{ m/s}\), \(v_i = 6.94 \text{ m/s}\), and \(t = 12 \text{ s}\), we calculate \(a = \frac{26.39 - 6.94}{12} = 1.62 \,\text{m/s}^2\).

Step 3: Calculate Minimum Length of On-Ramp

Using the equation of motion \(d = v_i t + \frac{1}{2} a t^2\) for the on-ramp, where \(v_i = 6.94 \text{ m/s}\), \(a = 1.62 \text{ m/s}^2\), and \(t = 12 \text{ s}\), we calculate:\[d = 6.94 \times 12 + \frac{1}{2} \times 1.62 \times (12)^2\]\[d = 83.28 + 116.64 = 199.92 \text{ m}\]

Step 4: Conclusion

The minimum length of the on-ramp required for a smooth transition from 25 km/h to 95 km/h in 12 seconds is approximately 200 meters.

Key Concepts

Kinematics

Kinematics is the study of motion without considering the forces that cause it. In this type of physics problem, we are interested in how something moves rather than why it moves.

• It involves various parameters such as displacement, velocity, and acceleration.
• We often use kinematic equations, which are mathematical relationships involving these parameters to solve motion-related problems.

In the given exercise, we analyze a car's motion on an on-ramp using kinematics to determine the minimum length needed for it to accelerate to highway speeds.

Speed Conversion

Speed conversion is crucial when working with different unit systems. It's important to maintain consistency in units for precise calculations.

• The speeds of the car are initially provided in kilometers per hour (km/h) and need to be converted to meters per second (m/s).
• This conversion is done using the factor: \(1 ext{ km/h} = \frac{1000}{3600} ext{ m/s}\).

For example, the car's initial speed of 25 km/h is converted to roughly 6.94 m/s, and its final speed of 95 km/h converts to approximately 26.39 m/s. By converting speeds, we unify the units used in our calculations, ensuring accuracy.

Equations of Motion

Equations of motion are key tools in physics to describe the behavior of moving objects. They relate speed, time, acceleration, and displacement.

• One such equation is: \(d = v_i t + \frac{1}{2} a t^2\), which helps calculate the total distance moved under uniform acceleration.
• Here, \(d\) represents distance, \(v_i\) is the initial velocity, \(t\) signifies time, and \(a\) denotes acceleration.

In the exercise, this formula aids in calculating the necessary length of the on-ramp, allowing us to ensure the car achieves the desired speed safely.

Acceleration

Acceleration measures how quickly an object's velocity changes. It is defined as the rate of change of velocity per unit time.

• It not only considers changes in speed but also in direction.
• Acceleration can be calculated with the formula \(a = \frac{v_f - v_i}{t}\), where \(v_f\) is the final velocity, \(v_i\) is the initial velocity, and \(t\) is the time taken for the change.

By understanding acceleration, we can determine how quickly the car accelerates on the ramp. In this scenario, the calculated acceleration of 1.62 m/s² allows us to further determine the appropriate length for the on-ramp.