Question

glider's instantaneous velocity when it is halfway through a photogate in terms of time

Answer 1

To determine the instantaneous velocity of a glider as it passes through a photogate when it is halfway through, we need to consider how the photogate and its setup work in terms of time and distance.

Let's denote:
- \( d \): the total width of the photogate (the distance that the glider travels while passing through the photogate).
- \( v(t) \): the instantaneous velocity of the glider at time \( t \).
- \( t = 0 \): the time when the glider is at the entrance of the photogate.
- \( t = T \): the time when the glider exits the photogate.

To get the instantaneous velocity at the halfway point of the photogate, we consider the midpoint distance, which is \( d/2 \).

### Steps to Find Instantaneous Velocity

1. **Determine the relationship between distance, time, and velocity**:
The average speed \( v_{\text{avg}} \) of the glider while passing through the photogate can be expressed as:
\[
v_{\text{avg}} = \frac{d}{T}
\]

2. **Finding Time Halfway Through**:
If the glider is uniform in its motion, we can find the time \( t_h \) spent reaching the halfway point:
\[
t_h = \frac{T}{2}
\]

3. **Calculate Instantaneous Velocity**:
The instantaneous velocity at the midpoint of the photogate would also correspond to the average velocity if the motion was uniform:
\[
v_h = \frac{d/2}{t_h} = \frac{d/2}{T/2} = \frac{d}{T}
\]

Thus, the instantaneous velocity \( v(t) \) of the glider when it is halfway through the photogate, assuming uniform motion, can be expressed in terms of the total distance \( d \) and the total time \( T \):
\[
v(t) = \frac{d}{T}
\]

If the glider is not moving uniformly (i.e., if there is acceleration), then you'd need the specific position as a function of time to calculate the exact instantaneous velocity. In that case, you would differentiate the position function to find velocity at the desired time.