Drawing Velocity Change Vectors- Physics Tutorial

What Is a Velocity Change Vector?

A velocity change vector shows the difference between two velocity vectors. You calculate it as:

Δv = v_final − v_initial

That's it. The change in velocity isn't just about speed changing—direction changes count too. A car going 60 mph north that turns east while maintaining 60 mph still has a change in velocity. Physics doesn't care if you think that's unfair.

Why Students Screw This Up

Most people draw the wrong thing because they confuse "change" with "final." They draw the final velocity vector instead of the difference between two vectors. That's not a velocity change vector—that's just a velocity vector.

The confusion comes from thinking of velocity as a single thing. But velocity is a vector—it has magnitude and direction. Change in velocity is the vector difference, not a single vector.

The Two Methods for Drawing Velocity Change Vectors

Method 1: Tip-to-Tail Subtraction

To subtract vectors tip-to-tail:

• Draw the initial velocity vector (v_i)
• Draw the final velocity vector starting from the tip of the initial vector
• The velocity change vector goes from the tail of v_i to the tip of v_f

This method works but gets messy when vectors aren't conveniently placed.

Method 2: Tail-to-Tail (Reversal Method)

This is cleaner and what most textbooks prefer:

• Place both velocity vectors tail-to-tail at the same point
• Reverse the initial velocity vector (multiply by −1)
• Draw the reversed v_i and the regular v_f tip-to-tail
• The resultant vector is the velocity change

When you reverse a vector, you're flipping its direction 180°. That's the key insight that makes this work.

Comparison: Which Method Should You Use?

The component method breaks vectors into x and y parts. Calculate Δv_x and Δv_y separately, then combine them. It's more work but eliminates geometric errors.

Step-by-Step: Drawing the Velocity Change Vector

Let's say you have:

• v_i = 20 m/s at 0° (pointing right)
• v_f = 20 m/s at 90° (pointing up)

Step 1: Place both vectors tail-to-tail at a point.

Step 2: Reverse v_i. Since v_i points right, −v_i points left with the same length.

Step 3: Place the tip of −v_i at the origin, then draw v_f starting from there.

Step 4: Draw the resultant from the tail of v_f to the tip of −v_i.

The result? A vector pointing up-left, at 45° if the magnitudes are equal. The magnitude works out to about 28.3 m/s using the Pythagorean theorem.

Common Mistakes That Cost You Points

• Drawing the final velocity instead of the change — This is the #1 error. Stop doing this.
• Forgetting that direction matters — A 90° turn isn't "no change" just because speed stays constant.
• Scaling errors — Make sure your arrow lengths match the vector magnitudes.
• Not reversing the initial vector — If you're using the reversal method and skip this step, everything after is wrong.

When Velocity Change Matters

You're really using this in Newton's Second Law problems (F = ma, where a = Δv/Δt) and impulse-momentum calculations. The change in velocity directly gives you acceleration, which tells you the net force.

In circular motion, even if speed is constant, velocity changes constantly because direction changes. That means there's acceleration. The velocity change vector at any instant points toward the center of the circle.

Quick Reference: Drawing Rules

• Velocity change = v_f − v_i
• Use tail-to-tail method: reverse v_i, add to v_f
• Arrow length = magnitude
• Arrow direction = vector direction
• Resultant goes from tail of v_f to tip of reversed v_i

Practice with simple cases first. Draw v_i and v_f at different angles, reverse one, add them. Check your work by calculating the magnitude and angle of the result.

Once you get the geometry down, physics problems become much easier. The hard part isn't the math—it's visualizing what the vectors actually represent.