Question

The elapsed time for a top fuel dragster to start from rest and travel in a straight line a distance of \(\frac{1}{4}\) mile \((402 \mathrm{~m})\) is \(4.441 \mathrm{~s}\). Find the minimum coefficient of friction between the tires and the track needed to achieve this result. (Note that the minimum coefficient of friction is found from the simplifying assumption that the dragster accelerates with constant acceleration. For this problem we neglect the downward forces from spoilers and the exhaust pipes.)

Short answer

Solving for acceleration, we get: $$ a = \frac{2 * 402}{4.441^2} = \frac{804}{19.722481} \approx 40.76 \frac{m}{s^2} $$ #tag_title# Step 2: Find the force required #tag_content# To find the force required, we can use Newton's second law: $$ F = ma $$ Where \(F\) is the force, \(m\) is the mass of the dragster, and \(a\) is the acceleration we found in step 1. However, we do not have the mass of the dragster given in the problem, so we cannot calculate the force directly. #tag_title# Step 3: Find the minimum coefficient of friction #tag_content# Since we do not have the mass of the dragster, we cannot calculate the force required directly. However, we are asked to find the minimum coefficient of friction between the tires and the track, which we can represent as: $$ \mu \geq \frac{F}{N} $$ Where \(\mu\) is the coefficient of friction, \(F\) is the force required (or the force exerted by the tires), and \(N\) is the normal force (equal to the weight of the dragster, which is the mass times the gravitational acceleration). Since the force required can be expressed as \(F = ma\), we can substitute this into the inequality: $$ \mu \geq \frac{ma}{mg} $$ Here, we see that the mass of the dragster will cancel out, allowing us to solve for the minimum coefficient of friction without knowing the mass: $$ \mu \geq \frac{a}{g} $$ Using the acceleration we calculated in step 1 (40.76 m/s²), and the gravitational acceleration (9.81 m/s²), we can find the minimum coefficient of friction: $$ \mu \geq \frac{40.76}{9.81} \approx 4.16 $$ Therefore, the minimum coefficient of friction needed for the top fuel dragster to achieve this result is 4.16.

Step-by-step solution

Find the acceleration

We will use the formula for distance in uniformly accelerated motion, which is: $$ d = v_0t + \frac{1}{2}at^2 $$ Where \(d\) is the distance, \(v_0\) is the initial velocity, \(t\) is the time and \(a\) is the acceleration. Since the dragster starts from rest, the initial velocity (\(v_0\)) is 0. Given the distance is \(\frac{1}{4}\) mile or 402 meters and the time is 4.441 seconds, we can solve for the acceleration (\(a\)): $$ 402 = 0 + \frac{1}{2}a(4.441^2) $$

Key concepts

Uniformly Accelerated Motion

Understanding uniformly accelerated motion is essential when studying the physics behind activities like drag racing. This type of motion occurs when an object is under constant acceleration, meaning that its velocity increases by the same amount every second. In the context of our problem, this translates into a dragster that starts from rest and speeds up steadily over a given distance.

For objects under uniformly accelerated motion, such as the dragster, we can apply specific kinematics equations to describe their motion. The equation we are considering in the provided exercise \[d = v_0t + \frac{1}{2}at^2\], where d represents the distance traveled, v_0 is the initial velocity, t is the time elapsed, and a is the constant acceleration, is perfect for situations where an object starts from rest and moves in a straight line, which is exactly what a top fuel dragster does during a race.

Using this formula allows us to solve for the dragster's acceleration during its quarter-mile sprint. This kinematic equation is fundamental because it relates acceleration directly to the distance traveled over a period of time, without needing the final velocity. It encapsulates the principles of uniformly accelerated motion, providing a clear path to determine the subsequent forces and coefficients at play, such as friction.

Physics of Drag Racing

Drag racing offers a vivid illustration of the principles of physics, particularly those of motion and friction. The sport is all about acceleration and speed, with cars like top fuel dragsters designed to cover a quarter mile from a standing start as quickly as possible.

In order to achieve maximum acceleration, the dragster must have an optimal interaction with the racing surface. This is where the coefficient of friction comes into play. The tires must grip the track firmly enough to convert the engine's power into forward motion without slipping. This grip is determined by the type of tire, the composition of the track surface, and the normal force—the weight of the dragster pressing down on the track.

The exercise in question aims to calculate the minimum coefficient of friction necessary for a dragster to achieve a certain acceleration without slip. This coefficient affects not only the start but the entire run, as it determines the maximum force that can be applied to the dragster in the direction of motion. Factors like the engine power, aerodynamics, drag, and the downforce from spoilers and exhaust pipes all contribute to the complexity of the physics involved in drag racing, although for simplicity, some of these factors are neglected in the initial problem formulation.

Kinematics Equations

The kinematics equations are powerful tools for analyzing motion in physics. They provide relationships between the position, velocity, acceleration, and time of a moving object, without the need to consider the forces involved. The equation used in the solution for the dragster's minimum coefficient of friction is one of these key kinematics equations.

This particular equation, often referred to as the second equation of motion, shows that if you know the initial velocity, time, and acceleration, you can calculate the distance traveled. Conversely, if you have the time, distance, and initial velocity, you can rearrange the equation to solve for acceleration, which is exactly what is done in the problem's step-by-step solution. Here is the equation again for reference:\[d = v_0t + \frac{1}{2}at^2\].

Because we're dealing with a scenario where the dragster is initially at rest (\(v_0 = 0\)), we simplify our equation further to \(d = \frac{1}{2}at^2\). By substituting the known values for d (the quarter-mile distance) and t (the elapsed time), we are able to isolate a and solve for the acceleration. With the acceleration known, other variables, such as the minimum coefficient of friction, can then be calculated using additional physics formulas that relate the forces at play with the kinematics of the dragster's motion.