A glider of length ℓmoves through a stationary photogate on...

A glider of length $\ell$ moves through a stationary photogate on an air track. A photogate (Fig. $\mathrm{P} 2.19$ ) is a device that measures the time interval $\Delta t_{d}$ during which the glider blocks a beam of infrared light passing across the photogate. The ratio $v_{d}=\ell / \Delta t_{d}$ is the average velocity of the glider over this part of its motion. Suppose the glider moves with constant acceleration. (a) Argue for or against the idea that $v_{d}$ is equal to the instantaneous velocity of the glider when it is halfway through the photogate in space. (b) Argue for or against the idea that $v_{d}$ is equal to the instantaneous velocity of the glider when it is halfway through the photogate in time.

Key Concepts

Constant Acceleration

In a uniformly accelerated motion, the acceleration remains fixed over time, meaning the velocity increases linearly. This linearity leads to a direct relation between the initial and final velocities over an interval, making the average velocity equal to the arithmetic mean of these velocities.

Average Velocity

The average velocity over a time interval is defined as the total displacement divided by the total time taken. In constant acceleration scenarios, this average is significant because it precisely equals the instantaneous velocity at the temporal midpoint of the interval, even though the same does not necessarily hold for the spatial midpoint.

Instantaneous Velocity

Instantaneous velocity is the speed of an object at a specific moment, reflecting the object's actual speed at that precise time. In contexts of constant acceleration, because the velocity changes linearly, measuring at a specific time or location can yield different insights about the object's motion, highlighting the difference between instantaneous and averaged quantities.

Midpoint Considerations in Uniformly Accelerated Motion

Under constant acceleration, while the average velocity over an interval equals the instantaneous velocity at the time midpoint, it does not generally equal the instantaneous velocity when the object’s position is halfway through the interval. This distinction is crucial in analyzing measurements that refer to spatial midpoints versus temporal midpoints.

Transcript

00:01 Okay, we have question 2 .19 and we have a glider moving through a photo gate.

00:11 And if you've ever used a photo gate, you know that it measures the time required, or the time that like a beam is blocked.

00:23 It uses that and the length of the object to figure out velocity, right? so they give us, as i say, that the average velocity vd of the glider is the length of the glider divided by the time it takes for the glider to pass through the gate or the time the beam in the photo gate is blocked, right? that's delta td.

00:52 So we suppose that acceleration is constant.

01:01 And then we want to think about is vd.

01:06 Equal to the instantaneous velocity of the glider when it's halfway through the photo gate in space.

01:13 So halfway through the photo gate in space means we'll do kind of like a top -down view.

01:20 Here's the photo gate, do the glider in red.

01:23 Here's your glider.

01:27 This is kind of a top -down view, right? our glider is moving, let's say it's moving to the right.

01:40 So it's halfway through the photo gate in space, meaning its length, if you found half of its length, half of its length would be right at the center of the photo gain.

01:54 So is vd equal to the instantaneous velocity of the glider at this point in space? the answer is no, because the acceleration, acceleration means the velocity, changes uniformly in time.

02:41 And specifically constant acceleration means this...

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