Question

A double speed trap is set up on a freeway. One police cruiser is hidden behind a billboard, and another is some distance away under a bridge. As a sedan passes by the first cruiser, its speed is measured to be \(105.9 \mathrm{mph}\). Since the driver has a radar detector, he is alerted to the fact that his speed has been measured, and he tries to slow his car down gradually without stepping on the brakes and alerting the police that he knew he was going too fast. Just taking the foot off the gas leads to a constant deceleration. Exactly 7.05 s later the sedan passes the second police cruiser. Now its speed is measured to be only \(67.1 \mathrm{mph}\), just below the local freeway speed limit. a) What is the value of the deceleration? b) How far apart are the two cruisers?

Short answer

Answer: The deceleration of the car is approximately -2.469 m/s², and the distance between the two police cruisers is approximately 232.177 meters.

Step-by-step solution

Convert speeds to m/s

First, we need to convert the speeds from mph to m/s. We can use the conversion factor: 1 mph = 0.44704 m/s. So, \(v_1 = 105.9 \, mph \times 0.44704 \frac{m}{s \cdot mph} \approx 47.339 \, m/s\) and \(v_2 = 67.1 \, mph \times 0.44704 \frac{m}{s \cdot mph} \approx 29.967 \, m/s\)

Calculate the deceleration

We can use the equation \(v_2 = v_1 + at\) to calculate the deceleration. Rearrange the equation to solve for 'a': \(a = \frac{v_2 - v_1}{t}\). Plug in the values we found in Step 1: \(a = \frac{29.967 \, m/s - 47.339 \, m/s}{7.05 \, s} \approx -2.469 \, m/s^2\)

Calculate the distance between the cruisers

We can use the equation \(s = v_1t + \frac{1}{2}at^2\) to find the distance between the cruisers. Plug in the values we found in the previous steps: \(s \approx 47.339 \, m/s \cdot 7.05 \, s - \frac{1}{2} \cdot 2.469 \, m/s^2 \cdot (7.05 \, s)^2 \approx 232.177 \, m\) The deceleration is approximately -2.469 \(m/s^2\), and the distance between the two cruisers is approximately 232.177 meters.

Key concepts

Deceleration Calculation

Deceleration refers to a decrease in the velocity of an object over time, and it's a crucial concept in physics, particularly in kinematics—the study of motion without considering its causes. When an object slows down, it experiences a negative acceleration, or deceleration. To calculate this uniformly changing motion, a specific kinematic formula is used:
\[\begin{equation}a = \frac{{v_{2} - v_{1}}}{{t}}\end{equation}\]Where:

• \[\[\begin{align*}a\end{align*}\]\] is the deceleration (measured in meters per second squared, m/s²).
• \[\[\begin{align*}v_{1}\end{align*}\]\] is the initial velocity (how fast the object is moving at the start).
• \[\[\begin{align*}v_{2}\end{align*}\]\] is the final velocity (the object's speed when measurement ends).
• t is the time over which the change in velocity occurs.

In our example, the sedan's speed decreases from 105.9 mph to 67.1 mph over 7.05 seconds. The negative sign in the deceleration value indicates that the velocity is decreasing. The accurate comprehension of deceleration is pivotal for students aiming to solve problems involving slowing down of objects—making it an essential skill for those studying physics.

Unit Conversion

Unit conversion is an essential skill in physics to ensure that all the measurements are in compatible units for calculation purposes. For example, in the case of our exercise, velocity units need to be converted from miles per hour (mph) to meters per second (m/s) since standard SI units are required for kinematic equations. Multiplying the given velocity by the conversion factor 0.44704, which corresponds to how many meters per second are in one mile per hour, facilitates a successful transition to the required unit system. Understanding and properly applying unit conversion avoids calculation errors and offers a robust foundation for various scientific disciplines, as well as everyday life instances where different measurement systems are frequently used.

Kinematic Equations

Kinematic equations are mathematical equations that describe the motion of objects without accounting for the forces that cause the motion. These equations are the heart of solving physics problems dealing with motion. There are several key kinematic equations, which include formulas to calculate displacement (s), velocity (v), acceleration (a), and time (t). A common one used in our scenario is:
\[\begin{equation}s = v_{1}t + \frac{1}{2}at^{2}\end{equation}\]This equation allows us to determine the distance traveled by the sedan between the two cruisers. Knowing how to manipulate and apply these equations is vital for students studying physics as they provide a systematic approach to analyzing motion, thus leading to a deeper understanding of the physical world. When used correctly, these equations serve as reliable tools for predicting future motion based on current conditions.