Instantaneous Velocity (2-3) - Key Takeaways
2-3 Instantaneous Velocity
Key concept: velocity describes how position changes with time; instantaneous velocity is the velocity at a specific instant, defined as the limit of the average velocity as the time interval goes to zero.
Notation:
Instantaneous velocity: v=
abla v_{ ext{instant}} = oxed{v= rac{\Delta x}{\Delta t}
abla\text{ in the limit }\Delta t\to 0}Average velocity (over a finite interval) is denoted with a bar: \bar{v}=\frac{\Delta x}{\Delta t}
One-dimensional motion sign convention:
Positive direction: typically to the right; negative to the left.
Important points:
If velocity is constant during an interval, the instantaneous velocity at any instant equals the average velocity over that interval.
If velocity changes, instantaneous velocity at a moment can differ from the average velocity over the interval.
The magnitude of instantaneous velocity equals instantaneous speed (since speed is the magnitude of velocity).
How graphs help: velocity as a function of time can show how instantaneous velocity relates to the average velocity (the average velocity is a global measure over a time interval; the instantaneous velocity is the slope-related concept at a precise moment).
Example relationships:
If a car moves 150 km in 2.0 h with varying speed, the average velocity over the trip is not simply the midpoint of speeds; the instantaneous velocity at any specific time is given by the limit definition above.
EXAMPLE 2-2 Distance a cyclist travels
Problem: How far can a cyclist travel in 2.5 h along a straight road if her average velocity is 18 km/h?
Approach: Use the definition of average velocity to solve for displacement Ax.
Formula:
Average velocity (Eq. 2-2): \bar{v}=\frac{Ax}{At}
Solve for displacement: Ax=v\,At
Solution: A_x=(18\ \text{km/h})(2.5\ \text{h})=45\ \text{km}.
EXAMPLE 2-3 Car changes speed
Problem: A car travels at a constant 50 km/h for 100 km. It then speeds up to 100 km/h and travels another 100 km. What is the car's average speed for the 200-km trip?
Approach: Compute times for each leg then use the definition of average velocity (Eq. 2-2).
Times:
At 50 km/h for 100 km: time = (100\,\text{km} / 50\,\text{km/h} = 2.0\,\text{h})
At 100 km/h for 100 km: time = (100\,\text{km} / 100\,\text{km/h} = 1.0\,\text{h})
Total distance: 200 km; total time: 3.0 h.
Average velocity (Eq. 2-2):
(\bar{v}=\dfrac{200\ \text{km}}{3.0\ \text{h}}\approx 66.7\ \text{km/h})
Important note (the conceptual pitfall): Averaging the two speeds gives (\frac{50+100}{2}=75\ \text{km/h}), which is not equal to the true average velocity. The correct approach uses the definition of average velocity, not the simple arithmetic mean of speeds.
Clarification: This demonstrates why you must use the definition of average velocity, not a naïve average of speeds.
2-3 Instantaneous Velocity (continued)
In a scenario where a car travels 150 km in 0.50 h with changing speed, the average velocity is
(\bar{v}=\dfrac{\Delta x}{\Delta t}=\dfrac{150\ \text{km}}{0.50\ \text{h}}=300\ \text{km/h}) (this is the average over that interval; instantaneous velocity at any instant would be given by the limit definition).
The instantaneous velocity is the velocity at a single moment, not an average over a span of time; the average velocity is a separate quantity defined over an interval.
EXERCISE B: Instantaneous speed at the turning point
Question: What is your instantaneous speed at the instant you turn around to move in the opposite direction?
Choices: (a) Depends on how quickly you turn around; (b) always zero; (c) always negative; (d) none of the above.
Correct answer: (b) always zero. At the exact turning point, the velocity is zero, so the instantaneous speed (the magnitude of velocity) is zero.
2-3 Key takeaways
The instantaneous velocity is the limit of the average velocity as the time interval goes to zero: $$v=\