Problem 122 - University Calculus: Early Transcendentals

Question

The State of Illinois Cycle Rider Safety Program requires motorcycle 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 approximately \(-21.51 \text{ ft/sec}^2\).

Step 1: Understand the Problem

We need to determine the constant deceleration required to bring a motorcycle to a stop from an initial speed of 44 ft/sec over a distance of 45 ft. This involves finding the acceleration that causes this change in speed over the given distance.

Step 2: Identify the Known Values

We know the initial velocity (\(v_i = 44 \text{ ft/sec}\)), the final velocity (\(v_f = 0 \text{ ft/sec}\)), and the distance over which this change occurs (\(d = 45 \text{ ft}\)). The deceleration needs to be a constant value.

Step 3: Use the Kinematic Equation for Constant Acceleration

The kinematic equation that relates initial velocity, final velocity, acceleration, and distance is given by:\[v_f^2 = v_i^2 + 2a d\]We rearrange this equation to solve for the acceleration (deceleration in this case, which will be a negative value):\[0 = (44)^2 + 2a(45)\]

Step 4: Solve for Deceleration

First, calculate \(44^2\): \[44^2 = 1936\]Plug in the values and solve for \(a\):\[0 = 1936 + 90a\]\[90a = -1936\]\[a = -\frac{1936}{90}\]\[a \approx -21.51 \text{ ft/sec}^2\]

Step 5: Interpret the Result

The negative sign in the acceleration value indicates deceleration. Thus, the constant deceleration required is approximately \(-21.51 \text{ ft/sec}^2\).

Key Concepts

Kinematic Equations

Kinematic equations are a critical part of physics that describe motion. They allow us to calculate various aspects of motion, such as velocity, acceleration, and distance, without needing time information explicitly. There are several kinematic equations, but one of the most important is:

• \[ v_f^2 = v_i^2 + 2a d \]

This particular equation is used when you need to find final velocity (\(v_f\)) or one of the other variables, such as distance (\(d\)) or acceleration (\(a\)), when time is not known. It's especially useful in situations involving constant acceleration.
In the context of our motorcycle problem, we use this equation to find the needed constant deceleration while braking. The equation helps us relate the initial speed, final speed, and the distance covered to calculate the constant acceleration (or deceleration). Remeber, when braking, the acceleration is negative, showing it's slowing down.

Velocity

Velocity is the speed of an object in a particular direction. It's a vector quantity, meaning it has both magnitude and direction.
For instance, a motorcycle traveling at 44 ft/sec to the east has a different velocity than one traveling 44 ft/sec to the west. The numerical value might be the same, but the directions are different.
In our motorcycle example, the initial velocity \(v_i\) is 44 ft/sec. This is how fast the motorcycle is going before the brakes are applied.

• Initial velocity: speed before deceleration begins.
• Final velocity: speed at the end of deceleration -- in this case, 0, since the motorcycle comes to a stop.

Understanding velocity is key to solving motion problems, as it helps us figure out how much change is required to reach a given state, like a complete stop.

Constant Acceleration

Constant acceleration refers to a consistent change in velocity over time. It means that the speed of an object either increases or decreases by the same amount each second.

• If acceleration is positive, the speed of the object is increasing.
• If acceleration is negative, it is decreasing, which we call deceleration.

In our problem, we calculate a constant deceleration to bring a motorcycle from 44 ft/sec to 0.
Using the kinematic equation, we calculated this deceleration as \(a \approx -21.51 \text{ ft/sec}^2\).
The negative sign indicates the motorcycle is slowing down. Constant acceleration simplifies calculations because we don't have to worry about changing rates, just a smooth change in velocity.

Distance

Distance is a measure of how far an object has traveled from its starting point. In physics, it is often represented as \(d\) in equations and calculations.
The distance an object covers can be when moving with a particular velocity, especially when under constant acceleration. In our motorcycle braking problem, the distance is 45 ft, the length over which the motorcycle comes to a full stop.

• Knowing the total distance is crucial when using kinematic equations, as it helps determine other variables like acceleration.
• In this context, distance explains the complete path the motorcycle takes in its process of slowing down.

When formulating and solving problems where motion is discussed, understanding the exact distance over which changes occur is essential, providing insight into how velocity and acceleration interact over space.