Topic 1.5 Notes – Vectors and Motion in Two Dimensions

1. What a Vector Is and How It Breaks into Components

A vector has both magnitude (how much) and direction (which way). Examples you see constantly in AP Physics 1:

• Displacement
• Velocity
• Acceleration
• Force

In two dimensions, any vector can be represented as the resultant of two perpendicular components, usually along the x- and y-axes.

You choose a coordinate system. Most of the time:

• +x → horizontal right
• +y → vertical up

That choice determines your signs.

Resolving a Vector into Components

If a vector A has magnitude $A$ and makes an angle $\theta$ measured from the +x axis:

$$A_x = A \cos \theta$$

$$A_y = A \sin \theta$$

These come straight from right-triangle trig:

• $\cos \theta = \frac{\text{adjacent}}{\text{hypotenuse}}$
• $\sin \theta = \frac{\text{opposite}}{\text{hypotenuse}}$

And the components relate back through the Pythagorean theorem:

$$A^2 = A_x^2 + A_y^2$$

Geometrically, you can think of the vector as the hypotenuse of a right triangle, with $A_x$ and $A_y$ as the legs:

Quick example:
A 12 m/s velocity at 30° above horizontal:

• $v_x = 12 \cos 30^\circ \approx 10.4 \text{ m/s}$
• $v_y = 12 \sin 30^\circ = 6.0 \text{ m/s}$

Rebuilding the Resultant from Components

If instead you’re given $A_x$ and $A_y$:

$$A = \sqrt{A_x^2 + A_y^2}$$

$$\theta = \tan^{-1}\left(\frac{A_y}{A_x}\right)$$

Be careful with signs and quadrants. If $A_x$ is negative, your angle is not in the first quadrant. On tests, students often get the magnitude right and the direction wrong.

The key idea: once broken into components, x and y behave independently. That’s what makes 2D motion manageable.

2. Motion in Two Dimensions Means Two Independent 1D Motions

Any object moving in a plane is just doing two 1D motions at the same time.

You apply the regular kinematics equations separately in x and y:

$$v = v_0 + at$$

$$x = x_0 + v_0 t + \frac{1}{2} a t^2$$

$$v^2 = v_0^2 + 2a(x - x_0)$$

The process always goes like this:

1. Break initial velocity into $v_{0x}$ and $v_{0y}$.
2. Identify $a_x$ and $a_y$.
3. Use the equations separately.
4. Remember time $t$ is the same in both directions.

The independence of x and y is one of the most tested ideas in this unit.

3. Projectile Motion

Projectile motion is a special case of 2D motion where:

• $a_x = 0$
• $a_y = -g \approx -9.8 \text{ m/s}^2$

No air resistance. Gravity is the only force.

Horizontal Motion

• Constant velocity
• $x = v_{0x} t$

There is no horizontal acceleration.

Vertical Motion

• Constant acceleration downward
• Treated exactly like free fall
• At the highest point, $v_y = 0$

Here’s the full picture with velocity components, maximum height, and range labeled:

Projectile motion with velocity components, maximum height (H), and range (R)

Horizontal vs Vertical Comparison

If the projectile lands at the same height it was launched:

• Time up = time down
• Range depends on both $v_{0x}$ and time in air
• Maximum height depends only on $v_{0y}$

A common AP-style conceptual question asks what happens to range if launch angle changes but speed stays the same. You should know that complementary angles like 30° and 60° give the same range on level ground.