Problem 96 - Thomas Calculus

Question

Stopping a motorcycle The State of Illinois Cycle Rider Safety Program requires riders to be able to brake from 30 mph (44 \(\mathrm{ft} / \mathrm{sec} )\) to 0 in 45 \(\mathrm{ft}\) . What constant deceleration does it take to do that?

Step-by-Step Solution

Answer

The constant deceleration is -21.51 ft/s².

Step 1: Understand the Problem Statement

We need to find the constant deceleration required to stop a motorcycle from an initial velocity of 30 mph (44 ft/s) to 0 over a distance of 45 ft.

Step 2: Establish the Known Variables

The initial velocity \( v_i \) is 44 ft/s, the final velocity \( v_f \) is 0 ft/s, and the stopping distance \( d \) is 45 ft.

Step 3: Apply the Kinematic Equation

Use the kinematic equation \( v_f^2 = v_i^2 + 2a d \), where \( a \) is the acceleration (deceleration in this case). Rearrange to solve for \( a \): \[ a = \frac{v_f^2 - v_i^2}{2d} \]

Step 4: Substitute the Known Values

Substitute the known values into the equation:\[ a = \frac{0^2 - 44^2}{2 \times 45} = \frac{-1936}{90} \]

Step 5: Calculate the Deceleration

Calculate the deceleration:\[ a = -21.51 \text{ ft/s}^2 \]

Key Concepts

Constant Deceleration

Constant deceleration refers to the rate at which an object's velocity decreases uniformly over time. When it comes to motion, deceleration is simply negative acceleration, meaning the object slows down rather than speeds up. In the context of braking, such as in the motorcycle stopping example, constant deceleration plays a critical role in safely bringing a vehicle to a stop.

To achieve constant deceleration, a rider must apply a steady force on the brakes, meaning the motorcycle's speed decreases evenly until it reaches zero velocity. This ensures smooth and controlled stopping. The magnitude of the deceleration depends on several factors including initial speed, braking force, and stopping distance.

Understanding how to calculate this parameter is crucial for assessing safety requirements and performance capability of vehicles, particularly in driver training programs where safety is paramount.

Motion Equations

Motion equations, often referred to as kinematic equations, are mathematical formulas used to describe the motion of objects. These equations can predict future motion and are essential in solving physics problems related to moving objects. Specifically, they allow us to relate velocity, acceleration, time, and distance.

In the exercise given, the relevant kinematic equation is:

• \[ v_f^2 = v_i^2 + 2a d \]

This equation is particularly useful because it connects an object's final velocity \( v_f \), initial velocity \( v_i \), acceleration \( a \) (or deceleration when negative), and the distance traveled \( d \). In this example, substituting the known values provides a way to solve for the required constant deceleration.

Understanding and applying motion equations is a fundamental skill in physics and engineering, as it allows for accurate predictions and insights into how and why objects move the way they do. Practicing these calculations helps reinforce theoretical understanding with practical applications.

Velocity Calculations

Velocity is a vector quantity that refers to the rate of change of an object's position with respect to time. It has both magnitude and direction, making it distinct from speed, which lacks directional information. Calculating velocity is a critical part of analyzing motion.

In the context of the motorcycle exercise, the initial velocity is 44 ft/s, and the final velocity is 0 ft/s, which are crucial for determining the deceleration. These values are plugged into the kinematic equation to find out how quickly the motorcycle needs to slow down to stop over a given distance.

This calculation can be broken down into steps:

• Identify known velocities and displacements.
• Use the appropriate motion equation for the scenario.
• Solve for the unknown variable, such as acceleration or time.

By mastering how to calculate velocity, students can gain a deeper understanding of motion dynamics, improve problem-solving skills, and apply these techniques across various real-world scenarios, enhancing both safety and efficiency in mechanical and transportation systems.