Problem 31 - Calculus Single Variable

Question

A car traveling at a speed of \(50 \mathrm{mi} / \mathrm{hr}\) must come to a halt in \(1200 \mathrm{ft}\). If the vehicle will decelerate at a constant rate, what should that rate be?

Step-by-Step Solution

Answer

The car should decelerate at a rate of -2.24 feet/second².

Step 1: Convert Speed to Feet per Second

The speed of the car is given in miles per hour (mph), which we need to convert to feet per second (fps). Use the conversion factors: 1 mile = 5280 feet and 1 hour = 3600 seconds. So, the speed in fps is calculated as: \[ 50 \text{ miles/hour} = 50 \times \frac{5280 \text{ feet}}{1 \text{ mile}} \times \frac{1 \text{ hour}}{3600 \text{ seconds}} = 73.33 \text{ feet/second} \]

Step 2: Apply the Kinematic Equation

We need to find the constant deceleration rate. We can use the kinematic equation:\[ v^2 = u^2 + 2a s \]where \(v\) is the final velocity (0 feet per second since the car stops), \(u\) is the initial velocity (73.33 feet per second), \(a\) is the acceleration (deceleration, so it will be negative), and \(s\) is the distance over which the deceleration occurs (1200 feet).

Step 3: Solve for Deceleration Rate

Insert the known values into the equation:\[0 = (73.33)^2 + 2 \times a \times 1200 \]Simplify and solve for \(a\):\[0 = 5377.78 + 2400a \]Rearrange to find \(a\):\[2400a = -5377.78 \]\[a = -\frac{5377.78}{2400} = -2.24 \text{ feet/second}^2\]

Step 4: Interpret the Result

The negative sign indicates that the car is decelerating. Therefore, the car must decelerate at a constant rate of \(-2.24 \text{ feet/second}^2\) to come to a stop in 1200 feet.

Key Concepts

Constant Deceleration

Deceleration refers to the reduction in speed over time. In physics, when deceleration is at a constant rate, it means the speed decreases uniformly over every second. This uniformity is particularly useful as it allows us to use kinematic equations to calculate important values like distance, speed, and time.

In the scenario of a car coming to a stop, constant deceleration implies that every second, the vehicle loses the same amount of speed until it halts.

• This is often described by a negative acceleration value, indicating a decrease in speed.
• The kinematic equation used, \[ v^2 = u^2 + 2as \]can help determine the necessary deceleration rate.
• Here, the final velocity (\( v \) ) is zero since the car stops.

The formula reflects how the initial speed, distance, and deceleration rate are interlinked. Understanding constant deceleration is crucial, particularly in scenarios where safety and controlled stoppage are priorities, such as in vehicle braking systems.

Unit Conversion

Accurate calculations in physics often require converting units to ensure consistency across the variables involved. For instance, in this exercise, the initial speed was given in miles per hour (mph). Converting this to feet per second (fps) is crucial because the rest of the data (like distance) is in feet.

• Consider that \( 1 \text{ mile} = 5280 \text{ feet} \).
• Also, \( 1 \text{ hour} = 3600 \text{ seconds} \).
• Thus, converting involves multiplying the speed by these factors to switch from mph to fps.

Unit conversion ensures that all measurements are in harmony in mathematical equations. Without aligned units, the calculations would be erroneous, leading to incorrect conclusions. So, mastering unit conversions is fundamental in solving physics problems efficiently and accurately.

Physics of Motion

The physics of motion encompasses the study of objects in movement, accounting for variables like velocity, acceleration (or deceleration), time, and distance. Understanding these factors is key to solving any motion-related problem.

Motion analysis usually involves:

• Velocity, which is the speed with a directional component.
• Acceleration, representing the rate at which speed changes over time. When negative, it signifies deceleration.
• Kinematic equations offer a mathematical framework to analyze motion. They relate the main aspects of motion in a structured way.

Considering a car stopping, it's about understanding how initial speed, stopping distance, and deceleration interact. Kinematic studies can predict outcomes or determine what's needed for specific motion goals (like coming to a halt safely). By incorporating these principles, physics provides a systematic approach to understanding and resolving motion-based problems.