Question

An airtrack glider, \(8.0 \mathrm{cm}\) long, blocks light as it goes through a photocell gate. The glider is released from rest on a frictionless inclined track and the gate is positioned so that the glider has traveled \(96 \mathrm{cm}\) when it is in the middle of the gate. The timer gives a reading of 333 ms for the glider to pass through this gate. Friction is negligible. What is the acceleration (assumed constant) of the glider along the track? (IMAGE CANNOT COPY)

Short answer

The glider's acceleration is approximately \(0.1152 \text{ m/s}^2\).

Step-by-step solution

Understanding the Problem

The problem involves determining the acceleration of a glider moving along a frictionless track. We know its length, how far it travels to the gate, and the time it takes to completely pass through the gate.

Identify Known Values

Given: - Length of the glider: 8.0 cm - Distance from rest to the center of the gate: 96 cm - Time through the gate: 333 ms We need to convert these measurements: - Distance to meters: 0.96 m - Time in seconds: 0.333 sec.

Determine Total Travel Distance to End of Gate

Since the glider blocks the light when it is halfway through (which is 96 cm), the total travel distance to completely exit the gate is the initial 96 cm plus half the glider's length. Calculate: Total distance = 0.96 m + (0.08 m / 2) = 1.0 m.

Calculate Final Velocity at the Gate

Use the formula that relates distance, initial velocity, time, and final velocity: \[ v = \frac{2L}{t} \]where \( L \) is the length of the glider (0.08 m) and \( t \) is the time it takes to pass through the gate (0.333 sec). Calculate: \[ v = \frac{2 \times 0.08}{0.333} \approx 0.48 \text{ m/s} \]

Calculate Initial Velocity

Since the glider was at rest at the top of the incline, the initial velocity when it started from rest is 0 m/s (\( v_0 = 0 \text{ m/s} \)).

Apply Kinematic Equation to Find Acceleration

The kinematic equation that relates initial velocity \(v_0\), final velocity \(v\), acceleration \(a\), and distance \(d\) is: \[ v^2 = v_0^2 + 2ad \]With \( v = 0.48 \text{ m/s}, v_0 = 0 \text{ m/s}, \) and \( d = 0.96 \text{ m} + 0.04 \text{ m} = 1.0 \text{ m} \), substitute and solve for \(a\): \[ (0.48)^2 = 0 + 2a(1.0) \]\[ a = \frac{(0.48)^2}{2} = 0.1152 \text{ m/s}^2 \]

Complete the Solution

Thus, the acceleration of the glider along the incline is approximately \(0.1152 \text{ m/s}^2\).

Key concepts

Understanding Acceleration in Kinematics

Acceleration is a key concept in kinematics, the branch of physics focusing on the motion of objects. It describes how fast the velocity of an object changes over time. When we say an object is accelerating, we mean that it is either speeding up, slowing down, or changing direction. In problems like the glider on an inclined track, understanding acceleration helps us determine how quickly the glider's speed increases as it moves down the track.

Acceleration is typically measured in meters per second squared ( ext{m/s}^2 ). This unit tells us how much the velocity (speed in a specific direction) changes every second. In the original exercise, the acceleration of the glider needed to be determined as it travels downward along the frictionless track. Since the problem specifies the track is frictionless, we simplify our calculations as no forces like friction affect the motion.

The formula for constant acceleration is often expressed as:

•  a = rac{v_f - v_i}{t} , where  v_f  is the final velocity, and  v_i  is the initial velocity.
• For our exercise, because the glider starts from rest,  v_i = 0 , simplifying the calculation to essentially:  a = rac{v_f}{t} .

Recognizing how initial and final velocities are used in these equations forms the foundation for solving motion-related problems.

Kinematic Equations in Motion Analysis

Kinematic equations allow us to solve various motion scenarios by relating four central aspects of motion: displacement, initial velocity, final velocity, and acceleration over time. These equations assume the acceleration is constant, which is crucial in analyzing how objects like our glider move along their paths.

For instance, consider the kinematic equation applied in the exercise:

•  v^2 = v_0^2 + 2ad 

This equation connects the initial velocity ( v_0 ), final velocity ( v ), the constant acceleration ( a ), and the distance ( d ) traveled.

Let's break it down further:

• Since the glider starts from rest,  v_0  is zero, simplifying our calculation to:  v^2 = 2ad .
• By knowing the distance traveled and final velocity at the given point, we can compute the glider's acceleration.

Determining acceleration this way allows us to understand the increase in velocity over the journey.

Realizing how each element in the kinematic equations contributes to solving motion problems makes it easier to predict and analyze various real-world motion scenarios.

Frictionless Motion Simplification

Motion without friction simplifies calculations significantly because friction introduces an opposing force that decelerates an object. Understanding frictionless motion focuses solely on gravity and the object's initial conditions, like initial velocity and position.

In scenarios like the glider on an inclined track, frictionless motion ensures accurate predictions of how fast the glider speeds up as it descends. With no friction, acceleration along the incline depends only on gravitational pull and the incline's angle.

• This allows the use of kinematic equations straightforwardly without adjusting for extra forces like friction.
• It simplifies calculations, providing direct insight into the natural laws of motion.

When assessing real-world examples, complete frictionlessness is rare. However, special settings, like the airtrack used for the glider, get close enough to practically eliminate friction, allowing clearer insights into fundamental physics dynamics without external interference.
Recognizing the idealized conditions of frictionless motion lets students appreciate conceptual learning in physics, offering a foundational understanding before introducing more complex, real-world factors.