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Chapter 4: Problem 52

A car braked with a constant deceleration of \(16 \mathrm{ft} / \mathrm{s}^{2},\) pro- ducing skid marks measuring 200 \(\mathrm{ft}\) before coming to a stop. How fast was the car traveling when the brakes were first applied?

Short Answer

Expert verified
The car was traveling at 80 ft/s when the brakes were first applied.

Step by step solution

01

Understand the Problem and Gather Information

The car decelerates at a constant rate of \(16 \text{ ft/s}^2\) and comes to a stop after leaving skid marks of 200 ft. We need to find the initial speed of the car when the brakes were applied.
02

Identify the Relevant Formula

We need to use the formula for motion with constant acceleration (or deceleration): \[v^2 = u^2 + 2as\]where \(v\) is the final velocity (0 ft/s since the car comes to a stop), \(u\) is the initial velocity (what we are trying to find), \(a\) is the acceleration (-16 ft/s虏 in this case because it's deceleration), and \(s\) is the distance (200 ft).
03

Substitute Known Values into the Formula

Since the car stops, the final velocity \(v = 0\).Substituting the known values into the equation:\[0 = u^2 + 2(-16)(200)\]
04

Solve for Initial Velocity

Rearrange the equation to solve for \(u^2\):\[u^2 = -2(-16)(200)\]\[u^2 = 6400\]Solving for \(u\) gives \(u = \sqrt{6400}\).
05

Calculate the Initial Velocity

Calculate the square root to give:\[u = 80 \text{ ft/s}\]

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Key Concepts

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

Deceleration
Deceleration refers to the rate at which an object slows down. It is essentially negative acceleration. In our car braking scenario, the vehicle is experiencing a reduction in speed, meaning its velocity is decreasing over time. The deceleration rate here is provided as \(-16 \text{ ft/s}^2\).
This value is negative because it reduces the velocity of the car as time progresses. Unlike acceleration where an object's speed increases, deceleration focuses on how swiftly an object can come to rest.
Understanding deceleration is crucial in physics problems involving stopping distances and times. You can calculate how long it takes for a moving object to come to a complete stop by knowing its initial velocity and rate of deceleration.
Here are some key points:
  • Deceleration is represented as a negative acceleration in physics equations.
  • Acceleration and deceleration are both measured in the same units: distance per time squared (like \(\text{ft/s}^2\)).
  • It's a critical factor in determining stopping distances and times in practical scenarios like driving.
Kinematics
Kinematics is a branch of mechanics that describes the motion of objects without considering the forces that cause the motion. It provides formulas and equations such as \(v^2 = u^2 + 2as\) which we use to solve problems about how objects move.
This field of study is essential in scenarios where we need to describe an object's path, velocity, and acceleration or deceleration over time. In our car example, we employ kinematic equations to determine how fast the car was moving before the brakes were applied.
Components of kinematics you'll frequently encounter include:
  • Displacement (\(s\)): This is the distance over which the object moves.
  • Initial velocity (\(u\)): The speed of the object when it starts its motion.
  • Final velocity (\(v\)): The speed at which the object comes to rest or reaches another specified value.
  • Acceleration (\(a\)): The rate of change of velocity, positive for speeding up and negative for slowing down.
Using these variables, kinematics enables us to predict future motion with precision.
Initial Velocity
Initial velocity refers to the speed of an object at the moment it begins to undergo acceleration or deceleration. This is a crucial value in physics, as it serves as the starting point for various calculations involving motion.
In our car problem, the initial velocity \(u\) was unknown, and that's what we aimed to discover by using the appropriate kinematic equation. Knowing the initial velocity can help us predict how far an object will travel under constant acceleration or deceleration.
When solving physics problems for initial velocity:
  • Identify your known values such as distance (displacement), acceleration or deceleration, and final velocity.
  • Choose the right kinematic formula based on the data you have.
  • Solve for initial velocity using algebraic manipulation of the chosen formula.
This approach allows us to ascertain the starting speed of moving objects and further analyze their motion.

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