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Velocity

Velocity

The rate of change of the position of an object. For motion in one dimension, such as along the number line, velocity is a scalar. For motion in two dimensions or through three-dimensional space, velocity is a vector.

 

 

See also

Speed

Key Formula

v=ΔxΔt=xf−xitf−tiv = \frac{\Delta x}{\Delta t} = \frac{x_f - x_i}{t_f - t_i}
Where:
  • vv = Average velocity over the time interval
  • Δx\Delta x = Change in position (displacement)
  • xfx_f = Final position
  • xix_i = Initial position
  • tft_f = Final time
  • tit_i = Initial time

Worked Example

Problem: A car is at position 20 meters east of a starting point at time t = 2 seconds. At time t = 6 seconds, the car is at position 100 meters east of the same starting point. What is the car's average velocity?
Step 1: Identify the known values. The initial position is 20 m, the final position is 100 m, the initial time is 2 s, and the final time is 6 s.
xi=20 m,xf=100 m,ti=2 s,tf=6 sx_i = 20 \text{ m}, \quad x_f = 100 \text{ m}, \quad t_i = 2 \text{ s}, \quad t_f = 6 \text{ s}
Step 2: Calculate the displacement (change in position).
Δx=xf−xi=100−20=80 m\Delta x = x_f - x_i = 100 - 20 = 80 \text{ m}
Step 3: Calculate the time interval.
Δt=tf−ti=6−2=4 s\Delta t = t_f - t_i = 6 - 2 = 4 \text{ s}
Step 4: Divide displacement by time to find average velocity.
v=804=20 m/sv = \frac{80}{4} = 20 \text{ m/s}
Answer: The car's average velocity is 20 m/s in the eastward direction.

Another Example

Problem: A jogger runs 300 meters east and then turns around and runs 100 meters west. The entire trip takes 200 seconds. Find the jogger's average velocity and average speed.
Step 1: Find the displacement. The jogger ends up 300 − 100 = 200 meters east of the starting point.
Δx=300−100=200 m (east)\Delta x = 300 - 100 = 200 \text{ m (east)}
Step 2: Calculate average velocity using displacement divided by time.
v=200200=1 m/s (east)v = \frac{200}{200} = 1 \text{ m/s (east)}
Step 3: Calculate average speed using total distance divided by time. The total distance traveled is 300 + 100 = 400 m.
speed=400200=2 m/s\text{speed} = \frac{400}{200} = 2 \text{ m/s}
Answer: The average velocity is 1 m/s east, while the average speed is 2 m/s. This shows how velocity and speed can differ when direction changes.

Frequently Asked Questions

What is the difference between velocity and speed?
Speed measures only how fast something moves — it is always positive or zero. Velocity measures how fast something moves and in what direction, so it can be negative (indicating motion in the opposite direction). Speed equals the magnitude (absolute value) of velocity.
Can velocity be negative?
Yes. In one dimension, a negative velocity means the object is moving in the negative direction (for example, to the left on a number line or downward). The sign does not mean the object is slowing down; it indicates direction.

Velocity vs. Speed

Velocity includes direction and is calculated from displacement (change in position). Speed ignores direction and is calculated from total distance traveled. For motion in a straight line without reversals, speed and the magnitude of velocity are equal. When an object reverses direction, average speed will be greater than the magnitude of average velocity, because distance traveled exceeds the net displacement.

Why It Matters

Velocity is the foundational concept linking position and time in calculus and physics. In calculus, instantaneous velocity is defined as the derivative of position with respect to time. Understanding velocity is essential for analyzing motion in everything from simple word problems to engineering and navigation.

Common Mistakes

Mistake: Using total distance instead of displacement when calculating velocity.
Correction: Velocity depends on displacement — the straight-line change from start to finish — not the total path length. Total distance gives you speed, not velocity.
Mistake: Thinking negative velocity means an object is slowing down.
Correction: A negative velocity simply means the object moves in the negative direction. Slowing down is described by acceleration opposing the direction of motion, which is a separate concept.

Related Terms