Problem 52 - Algebra 2

Question

Olivia was driving her car at a speed of 65 miles per hour. To exit the highway, she began decelerating at a rate of 5 mph per second. How long did it take Olivia to come to a stop?

Step-by-Step Solution

Answer

It took approximately 13 seconds for Olivia to stop.

Step 1: Understanding the Problem

Olivia is driving at an initial speed of 65 miles per hour and decelerates at a rate of 5 miles per hour per second. We need to determine how long it will take for her to stop, meaning when her speed will reach 0 mph.

Step 2: Convert Units

First, convert the initial speed from miles per hour to miles per second. Since there are 3600 seconds in an hour, we calculate: \(65 \text{mph} = \frac{65}{3600} \text{ miles per second} \approx 0.01806 \text{ miles per second}\).

Step 3: Set Up the Equation

Let's use the equation for speed with constant acceleration: \(v = u + at\), where \(v\) is the final speed, \(u\) is the initial speed, \(a\) is the acceleration (negative for deceleration), and \(t\) is the time.Here, \(v = 0\), \(u = 65 \text{ mph}\) which equals approximately \(0.01806 \text{ miles per second}\), and \(a = -5 \text{ mph per second}\).

Step 4: Solve for Time

Rearrange the equation to solve for \(t\):\(0 = 0.01806 + (-5)t\).Simply, \(0 = 0.01806 - 5t\).Therefore, \(t = \frac{0.01806}{5} \approx 0.003612\) seconds.

Step 5: Convert Time to Appropriate Units

Since our calculation is in seconds due to how we set up the deceleration, no further conversion is necessary for this unit context. However, given the units transition and context, decimal points may offer further clarity.

Key Concepts

Unit Conversion

Unit conversion serves as a fundamental step in solving many physics problems, especially involving speed and acceleration. In this exercise, the initial speed was given in miles per hour, but the need arose to express it in miles per second to match the unit of deceleration, which is in miles per hour per second.

• Start by understanding the units: 1 hour equals 3600 seconds.
• To convert miles per hour to miles per second, divide the speed by 3600: \(65 \text{ mph} = \frac{65}{3600} \text{ miles per second} \approx 0.01806 \text{ miles per second}\).

This initial conversion is crucial because consistent units allow equations to be set and solved correctly. Small steps like these make sure calculations remain valid throughout.

Constant Acceleration Equation

The constant acceleration equation is a cornerstone of kinematics, used to describe the motion of objects under constant acceleration or deceleration. The equation used here is:\(v = u + at\)Where:

• \(v\) is the final velocity.
• \(u\) is the initial velocity.
• \(a\) is the constant acceleration, which is negative in case of deceleration.
• \(t\) is the time.

In Olivia's problem, the goal is to solve for time \(t\) when her final speed \(v\) becomes 0, using the known rates of initial speed \(u = 0.01806 \text{ miles per second}\) and deceleration \(a = -5 \text{ mph per second}\). This formula is a linchpin in solving problems regarding motion with constant acceleration.

Deceleration Rate

Understanding a deceleration rate helps determine how quickly an object reduces its speed. In this problem, Olivia's car decelerates at 5 miles per hour per second. Here's how it works:
The rate tells us how much the speed decreases every second:

• Every second, the speed reduces by 5 miles per hour.
• This consistent reduction helps build a clear picture of how *fast* a vehicle comes to a stop.

This concept acts as an intuition builder in kinematics, painting a real-world scenario where one can predict how quickly things slow down. Accurate understanding of a deceleration rate allows for the application of the constant acceleration equation to find additional motion characteristics like time.

Speed and Velocity

Speed and velocity, although used interchangeably in everyday language, have specific meanings in physics.
Speed is a scalar quantity that refers only to how fast an object is moving, irrespective of direction, measured in units like mph or m/s. Velocity is a vector quantity and includes both speed and direction.

• Olivia's initial speed was 65 mph, in some direction along the highway.
• For deceleration calculations, speed's absolute value without directional context was considered: the shift was towards zero velocity.

Understanding these two quantities is fundamental in solving motion problems, as it allows us to accurately predict and compute motion vectors and behaviors under acceleration or deceleration scenarios.