AP Physics 1: Vector Addition in Kinematics

Mastering vector addition is the key that unlocks two-dimensional motion in physics. Unlike simple arithmetic, adding vectors requires both magnitude and direction, a skill essential for predicting an object’s net displacement, final velocity, or overall acceleration.

What Makes a Vector Different from a Number?

In kinematics, you deal with two types of quantities. Scalar quantities are described by magnitude (size) alone. Examples include mass, temperature, time, and speed. Vector quantities, however, are described by both magnitude and direction. The core kinematic vectors are displacement (change in position), velocity (rate of change of displacement), and acceleration (rate of change of velocity).

The distinction is crucial. If you walk 3 meters east and then 4 meters north, your total distance (a scalar) is 7 meters. Your net displacement (a vector), however, is about 5 meters at an angle of 53° north of east. Adding vectors properly gives you this resultant, the single vector that has the same effect as all the individual vectors combined. Failing to account for direction is the most common beginner mistake in kinematics.

The Tip-to-Tail Graphical Method

The graphical method provides an intuitive, visual understanding of vector addition. To add two vectors graphically, you use the tip-to-tail method. You draw the first vector to scale. Then, you draw the second vector so that its "tail" starts at the "tip" (arrowhead) of the first vector. The order in which you add them does not matter; vector addition is commutative.

The resultant vector is then drawn from the tail of the first vector to the tip of the last vector. This new arrow represents the sum. For example, to find the net displacement of someone walking 25 m East (Vector A) and then 15 m North (Vector B), you would draw A, then draw B starting from A’s tip. The resultant, R, stretches from the start point to the end point. You can measure its length with a ruler (using your scale) to find the magnitude and use a protractor to find its direction relative to an axis.

While powerful for visualization, this method’s precision is limited by your drawing tools. It is, however, excellent for estimating and for understanding the concept of a resultant. On the AP exam, you might use it to sketch a problem, but for precise answers, you'll need an algebraic approach.

The Algebraic Component Method

For exact calculations, the component method is indispensable. This method breaks every vector down into perpendicular parts, usually along the x- (horizontal) and y- (vertical) axes. Resolving a vector into components is like finding out how much of it points purely east and how much points purely north.

For a vector A with magnitude A and direction θ measured from the positive x-axis:

• The x-component is $A_x=A\cos \theta$
• The y-component is $A_y=A\sin \theta$

Crucially, direction determines the sign (+ or -) of each component. East and North are typically positive; West and South are negative.

The power of this method shines when adding multiple vectors:

1. Resolve: Break all vectors into their x- and y-components.
2. Sum: Add all the x-components together to get the resultant’s x-component, $R_x$. Do the same for the y-components to get $R_y$.
3. Recombine: Use the Pythagorean theorem to find the magnitude of the resultant: $R=\sqrt{R_x^2+R_y^2}$.
4. Find Direction: Use the inverse tangent function to find the angle θ of the resultant relative to the x-axis: $\theta ={\tan }^{-1}(|R_y/R_x|)$. You must then use the signs of $R_x$ and $R_y$ to place the angle in the correct quadrant.

Worked Example: Finding Net Displacement

A drone flies 100 m at 30° North of East, then 80 m at 60° North of West. Find its net displacement.

Step 1: Resolve into Components.

• Vector 1 (100 m at 30° E of N): ${\theta }_1=30^{\circ }$
• $A_{1x}=(100)\cos (30^{\circ })=86.6$ m
• $A_{1y}=(100)\sin (30^{\circ })=50.0$ m
• Vector 2 (80 m at 60° N of W): The angle from the positive x-axis is $180^{\circ }-60^{\circ }=120^{\circ }$.
• $A_{2x}=(80)\cos (120^{\circ })=-40.0$ m
• $A_{2y}=(80)\sin (120^{\circ })=69.3$ m

Step 2: Sum the Components.

• $R_x=86.6+(-40.0)=46.6$ m
• $R_y=50.0+69.3=119.3$ m

Step 3 & 4: Recombine for Magnitude and Direction.

• Magnitude: $R=\sqrt{(46.6{)}^2+(119.3{)}^2}\approx 128$ m
• Direction: $\theta ={\tan }^{-1}(119.3/46.6)\approx 68.7^{\circ }$. Since $R_x$ > 0 and $R_y$ > 0, the resultant is in the first quadrant.
• Result: The net displacement is 128 m at 68.7° North of East.

Applying Vector Addition to Kinematics Problems

Vector addition is not just for displacements. The same rules apply identically to velocity and acceleration.

• Relative Velocity: A classic AP problem involves a boat crossing a river. The boat’s velocity relative to the water and the water’s velocity (current) relative to the Earth are two vectors. The boat’s resultant velocity relative to the Earth is the vector sum of these two. You use component addition to find how far downstream it lands.
• Projectile Motion: The motion of a projectile is analyzed by separating it into independent x- and y-components. The constant velocity in the x-direction and the uniformly accelerated motion in the y-direction are treated separately, and vector addition is used to recombine them to find the overall velocity at any point: $\vec{v}=v_x\hat{i}+v_y\hat{j}$.
• Net Force and Acceleration: While formally part of dynamics (Unit 2), the principle is identical. Newton’s second law, ${\vec{F}}_{net}=m\vec{a}$, is a vector equation. Finding the net acceleration from multiple forces is a vector addition problem, directly impacting how an object’s velocity will change.

Common Pitfalls

1. Adding Magnitudes Directly: The most fundamental error is treating vectors like scalars. If two velocity vectors are at an angle, their resultant magnitude is less than or equal to their numerical sum. It is only equal if they point in the exact same direction.
2. Component Sign Errors: Misidentifying the angle from the positive x-axis or forgetting that components can be negative will derail your calculation. Always sketch the vector quickly to check: if it points left, $A_x$ is negative; if it points down, $A_y$ is negative.
3. Incorrect Resultant Angle: Finding the angle using ${\tan }^{-1}(R_y/R_x)$ gives an answer between -90° and +90°. You must look at your $R_x$ and $R_y$ sums to determine the actual quadrant. For example, if $R_x$ is negative and $R_y$ is positive, the angle from the positive x-axis is $180^{\circ }-|\theta |$.
4. Confusing Graphical and Algebraic Methods: The graphical method is for visualization and estimation. On the AP exam’s free-response section, you must show your algebraic component work for full credit. A sketch is helpful, but it is not a substitute for calculation.

Summary

• Vectors have magnitude and direction; scalars have only magnitude. Displacement, velocity, and acceleration are the key kinematic vectors.
• The tip-to-tail graphical method visually demonstrates how to find a resultant vector, which represents the sum of all individual vectors.
• The algebraic component method is the precise, calculation-based technique. You resolve vectors into x- and y-components, sum the components separately, and then recombine them using the Pythagorean theorem and inverse tangent.
• Vector addition rules apply universally to displacement, velocity, and acceleration, forming the foundation for analyzing 2D motion, including relative velocity and projectile motion.
• Always check the quadrant of your resultant vector by examining the signs of its components, and never simply add vector magnitudes together.