Problem 100 - Thomas Calculus in SI Units

Question

Stopping a motorcycle The State of Illinois Cycle Rider Safety Program requires motorcycle riders to be able to brake from \(48 \mathrm{km} / \mathrm{h}(13.3 \mathrm{m} / \mathrm{s})\) to 0 in \(13.7 \mathrm{m} .\) What constant deceleration does it take to do that?

Step-by-Step Solution

Answer

The constant deceleration needed is approximately -6.46 m/s².

Step 1: Identify known values

We know the initial velocity \( v_i = 13.3 \, \text{m/s} \), the final velocity \( v_f = 0 \, \text{m/s} \), and the stopping distance \( d = 13.7 \, \text{m} \). We need to find the constant deceleration \( a \).

Step 2: Use the kinematic equation

We will use the equation \( v_f^2 = v_i^2 + 2a d \), where \( v_f \) is the final velocity, \( v_i \) is the initial velocity, \( a \) is the acceleration (or deceleration in this context), and \( d \) is the distance.

Step 3: Rearrange the kinematic equation

Rearrange the equation to solve for \( a \):\[ a = \frac{v_f^2 - v_i^2}{2d} \]

Step 4: Substitute known values into the equation

Substitute the known values into the equation:\[ a = \frac{0^2 - (13.3)^2}{2 \times 13.7} \]

Step 5: Calculate the deceleration

Calculate \( a \):\[ a = \frac{-176.89}{27.4} \approx -6.46 \, \text{m/s}^2 \]The negative sign indicates deceleration.

Key Concepts

Constant Deceleration

Deceleration refers to the process of slowing down an object in motion. Unlike acceleration, which speeds up an object, deceleration reduces its velocity over time. In physics, this is often referred to as negative acceleration because it has an opposite effect compared to speeding up.
In the exercise, the motorcycle rider must come to a complete stop from a speed of 13.3 m/s within a distance of 13.7 meters. The task is to determine the constant rate of deceleration needed to achieve this. Constant deceleration implies that the rate of slowing down remains the same throughout the motion until the vehicle stops.
To find this, we use the kinematic equation that relates velocity, acceleration, and distance to find the deceleration. Since the deceleration in this scenario is constant, the motion of the motorcycle can be calculated predictably using the equation provided.

Kinematic Equations

Kinematic equations are essential tools in physics that help describe an object's motion. They allow us to calculate different parameters like distance, speed, and acceleration. These equations assume that the motion involved occurs in a straight line and the acceleration is constant.
For our exercise, we used the equation:

• \[ v_f^2 = v_i^2 + 2a d \]

Where

• \( v_f \) is the final velocity,
• \( v_i \) is the initial velocity,
• \( a \) is the acceleration or deceleration,
• \( d \) is the distance.

By rearranging this equation, we can solve for the deceleration \( a \):\[ a = \frac{v_f^2 - v_i^2}{2d} \]This formula shows how the final and initial velocities, along with the distance, determine the rate of deceleration.

Motion in Physics

Motion in physics refers to a change in position of an object over time. It is one of the fundamental concepts that includes velocity, acceleration, and the trajectory of a moving object.
In this exercise, we specifically deal with linear motion, which is straightforward and occurs along a straight path. Understanding motion helps us explain how vehicles move and stop, which is crucial for safety applications like motorcycle braking scenarios.
Physics provides various quantitative methods through equations of motion that allow us to predict future movement based on present conditions. By examining initial conditions (like initial speed) and requirements (such as stopping distance), we can determine necessary adjustments to speed or acceleration, thus thoroughly understanding the entire scenario of motion stopping.