Question

The brakes on your automobile are capable of creating a deceleration of \(17 \mathrm{ft} / \mathrm{s}^{2} .\) If you are going \(85 \mathrm{mi} / \mathrm{h}\) and suddenly see a state trooper, what is the minimum time in which you can get your car under the \(55 \mathrm{mi} / \mathrm{h}\) speed limit?

Short answer

The minimum time in which the car can be under the 55 mi/h speed limit is approximately 2.59 seconds.

Step-by-step solution

Convert speeds to feet per second

First, convert the speeds from miles per hour to feet per second for consistency with the unit of deceleration. 1 mile is equal to 5280 feet and 1 hour is equal to 3600 seconds, so \( 85 \mathrm{mi/h} = 85 * (5280/3600) \mathrm{ft/s} \) and \(55 \mathrm{mi/h} = 55 * (5280/3600) \mathrm{ft/s}. \)

Calculate initial and final velocities

Next, calculate the initial and final velocities in feet per second. The initial velocity \(u\) is the speed of the car going at \(85 \mathrm{mi/h}\) and the final velocity \(v\) is the speed limit of \(55 \mathrm{mi/h}\). After calculating, we get \(u = 124.67 \mathrm{ft/s}\) and \(v = 80.67 \mathrm{ft/s}\).

Use the deceleration formula

The deceleration formula tells us that deceleration = (initial velocity - final velocity)/time. We can rearrange the formula to find time = (initial velocity - final velocity)/deceleration. Substitute the known values into the formula to find the time, \(t = (u-v)/a = (124.67-80.67)/17 \mathrm{s}\)

Key concepts

Uniform Deceleration

When discussing motion in physics, it's essential to understand the concept of uniform deceleration, which occurs when an object slows down at a steady rate. In our exercise case, a car is decelerating uniformly to reduce its speed in the presence of a state trooper.

Uniform deceleration can be visually represented on a velocity-time graph as a straight line with a negative slope, indicating a steady reduction in velocity over time. The slope of this line is equal to the deceleration of the object. Deceleration, in this context, is a vector quantity implying that it not only has magnitude but also direction, opposing the direction of motion.

This concept becomes particularly important in safety features of vehicles, like brakes, where the ability to decelerate uniformly allows for predictable and safe stopping distances.

Velocity Conversion

When working on physics problems, especially those involving motion, it may be necessary to convert velocities between different units of measure to ensure consistency and accuracy in calculations. In our exercise, the speeds were initially given in miles per hour (mi/h) but had to be converted to feet per second (ft/s) to match the unit of deceleration.

The conversion involves multiplying the speed value by a conversion factor that relates miles to feet and hours to seconds—specifically, the fact that there are 5280 feet in a mile and 3600 seconds in an hour. Understanding how to effectively convert between units is a critical skill in physics, as it allows for the appropriate application of formulas and principles across different scenarios.

Kinematics Equations

Kinematics is a branch of mechanics that describes the motion of objects without considering the causes of motion. Several kinematic equations enable us to relate variables like velocity, acceleration, displacement, and time. In our automotive deceleration scenario, we used one such equation, which is derived from the definition of acceleration (or in this case, deceleration): acceleration is the change in velocity over time.

By rearranging this definition, you can solve for the unknown time variable. Although we handled this without directly referencing displacement or the distance the car travels while slowing down, other kinematic equations include these variables and can be used for more complex calculations. Knowing which equation to use and how to manipulate it to solve for the desired variable is foundational in solving kinematic problems in physics.