Question

The fastest speed in NASCAR racing history was \(212.809 \mathrm{mph}\) (reached by Bill Elliott in 1987 at Talladega). If the race car decelerated from that speed at a rate of \(8.0 \mathrm{~m} / \mathrm{s}^{2},\) how far would it travel before coming to a stop?

Short answer

Answer: The car travels 566.675 meters before coming to a stop.

Step-by-step solution

Convert initial speed to m/s

To convert the initial speed from mph to m/s, we can use the conversion factor 1 mph = 0.44704 m/s. So, Initial speed in m/s = 212.809 mph * 0.44704 m/s / 1 mph = 95.104 m/s

Plug the values into the equation of motion

Using the equation v^2 = u^2 + 2as, we have: 0^2 = (95.104 m/s)^2 + 2 * -8.0 m/s² * s

Solve for the distance s

Now, we can solve the equation for the distance s: s = (0^2 - (95.104 m/s)^2) / 2 * -8.0 m/s² = (0 - 9046.8) / (-16) = 566.675 m So, the car would travel 566.675 meters before coming to a stop.

Key concepts

Kinematic Equations

Kinematic equations are the formulas that describe the motion of objects moving with constant acceleration. In the context of NASCAR racing, imagine a race car zooming around the track; these equations help us understand how the car's speed changes over time or distance.

For our NASCAR problem, we focus on the equation that relates velocity, acceleration, and displacement, often expressed as \( v^2 = u^2 + 2as \) where \( v \) is the final velocity, \( u \) is the initial velocity, \( a \) is acceleration (or deceleration, when it's negative), and \( s \) is the displacement or distance traveled. By using this equation, we can calculate how far the car traveled during its deceleration phase before coming to a stop.

Unit Conversion

Unit conversion is essential in physics problems because it ensures that all measurements are in consistent units, allowing us to correctly apply formulas. In our NASCAR physics problem, we started with a speed given in miles per hour (mph) but needed to convert it to meters per second (m/s), since meters per second is the standard unit for speed in the metric system, which is commonly used in physics.

We use the conversion factor \( 1 \text{ mph} = 0.44704 \text{ m/s} \) to switch from imperial to metric. Multiplying the speed in mph by this conversion factor gives us the speed in m/s, which we can then plug into our kinematic equation. Unit conversion like this is crucial in many physics problems and often one of the first steps to consider.

Deceleration

Deceleration is simply negative acceleration; it's the rate at which an object slows down. In the case of the NASCAR problem, the race car decelerates from a very high speed to a stop. It's important to note that when we work with deceleration in kinematic equations, we use a negative value for acceleration.

Here we're told that the deceleration is \( 8.0 \text{ m/s}^2 \) — remember, this is a positive number describing the rate of slowing, but since it reduces the velocity, we enter it as negative in our equation. Understanding deceleration is critical for accurately predicting how long it will take a moving object to come to a complete stop, given its initial velocity and rate of deceleration.

Motion Equations Physics

Motion equations in physics are mathematical representations of the principles of motion. They not only encompass kinematic equations but also other aspects, such as force, mass, and momentum. These equations allow us to predict future motion based on current conditions.

In simpler cases of motion with constant acceleration (or deceleration), such as our NASCAR example, we often use kinematic equations. These powerful tools in physics describe not only cars on a track but also everyday occurrences like a ball being thrown up in the air and various engineering applications. Understanding these equations lets you dissect complex motion into manageable portions that can be analyzed and predicted with precision.