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Chapter 1: Problem 31

The skid marks made by an automobile indicated that its brakes were fully applied for a distance of \(75 \mathrm{~m}\) before it came to a stop. The car in question is known to have a constant deceleration of \(20 \mathrm{~m} / \mathrm{s}^{2}\) under these conditions. How fast-in \(\mathrm{km} / \mathrm{h}\) -was the car traveling when the brakes were first applied?

Short Answer

Expert verified
The initial speed was approximately 197.16 km/h.

Step by step solution

01

Identify the known and unknown variables

We know the distance the car traveled while braking is \( d = 75 \) m, and the deceleration is \( a = -20 \text{ m/s}^2 \). The final velocity \( v_f \) is 0 m/s because the car comes to a stop. We need to find the initial velocity \( v_i \) in \( \text{km/h} \).
02

Use the kinematic equation

The kinematic equation without time is \[ v_f^2 = v_i^2 + 2ad \]. Substituting the known values \( v_f = 0 \), \( a = -20 \text{ m/s}^2 \), and \( d = 75 \text{ m} \), we have: \[ 0 = v_i^2 + 2(-20)(75) \].
03

Solve for initial velocity

Rearrange the equation \[ 0 = v_i^2 - 3000 \] to find \( v_i^2 = 3000 \). Taking the square root gives \( v_i = \sqrt{3000} \approx 54.77 \text{ m/s} \).
04

Convert to km/h

Convert the initial velocity from m/s to km/h by multiplying by \( 3.6 \). Thus, \( v_i = 54.77 \times 3.6 \approx 197.16 \text{ km/h} \).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Understanding Deceleration
Deceleration is the process of slowing down an object. In physics, it is often represented as a negative acceleration because it reduces an object's speed over time.
In the context of this exercise:
  • The car experiences a deceleration of \(-20 \, \text{m/s}^2\), relative to the force applied by the brakes.
  • This means for every second the brakes are applied, the speed of the car decreases by \(20 \, \text{m/s}\).
Deceleration plays a crucial role in controlling motion. It's the measure of an object's ability to come to a stop safely and can depend on factors such as road conditions and brake efficiency. Understanding deceleration allows engineers to design safety mechanisms in vehicles, ensuring they can stop within a certain distance.
Initial Velocity and Its Importance
The initial velocity is essentially how fast an object is moving at the start of observation. It gives us insight into the object's starting conditions before any forces, like braking forces, act on it.
In this exercise:
  • The car's initial velocity is determined just before the brakes are applied.
  • To solve for the initial velocity, we use data such as the distance and deceleration during braking.
  • The calculated initial velocity of the car was approximately \(54.77 \text{ m/s}\) or \(197.16 \text{ km/h}\).
Proper determination of initial velocity is crucial in many fields, including road safety assessments and accident investigations. It helps to reconstruct scenarios and assess if vehicles were operating within safe speed limits.
Essentials of Kinematic Equations
Kinematic equations are a backbone of motion analysis in physics. They help predict an object's position, velocity, and acceleration at any given time, assuming constant acceleration.
For this exercise, the applicable kinematic equation is:
  • Without time: \[ v_f^2 = v_i^2 + 2ad \] where:
    • \(v_f\) is the final velocity (0 \text{ m/s}, since the car stops),
    • \(v_i\) is the initial velocity,
    • \(a\) is the acceleration or deceleration (-20 \text{ m/s}^2 in this case),
    • \(d\) is the distance covered (75 \text{ m}).
By substituting the known quantities and rearranging the formula, we calculated the initial velocity, which was then converted to \( \text{km/h} \). Kinematic equations are powerful tools that provide a complete picture of an object's motion, crucial for engineering, sports science, and more.

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Most popular questions from this chapter

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