# Kinematics in Two Dimensions and Projectile Motion

Physics I · Principles of Mechanics · https://tryals.app/learn/physics-i/kinematics-in-two-dimensions-and-projectile-motion

## 2D Kinematics and Vectors

Fire a bullet horizontally and drop another simultaneously: they hit the ground together. Gravity is indifferent to horizontal motion, splitting every projectile problem into two independent stories sharing only the clock.

In 2D, position and velocity are **vectors** whose horizontal and vertical components act independently.

$$\vec{r}(t) = x(t)\hat{i} + y(t)\hat{j}, \quad \vec{v}(t) = \frac{d\vec{r}}{dt}$$

- The horizontal velocity $v_x = v_0\cos\theta$ is **constant** with zero acceleration.
- The vertical motion is **free fall** with $a_y = -g$ acting at all times.

> **Common pitfall:** At the peak, the projectile is **not** force-free. Its vertical velocity hits zero, but gravity still acts at full strength, pulling it back down.

## Projectile Formulas and Trajectory

For a projectile launched at speed $v_0$ and angle $\theta$, neglecting air resistance, the path is a **parabola** described by:

$$x(t) = v_0\cos\theta\,t, \quad y(t) = v_0\sin\theta\,t - \tfrac{1}{2}gt^2$$

| Quantity | Formula |
| --- | --- |
| **Time of flight** ($T$) | $T = \dfrac{2v_0\sin\theta}{g}$ |
| **Maximum height** ($H$) | $H = \dfrac{v_0^{2}\sin^{2}\theta}{2g}$ |
| **Range** ($R$) | $R = \dfrac{v_0^{2}\sin 2\theta}{g}$ |

- **Maximum range** occurs at $\theta = 45^\circ$ in a vacuum.
- **Complementary angles** ($\theta$ and $90^\circ - \theta$) yield the **same range**.

## Practice questions

3 of this lesson's 12 practice questions, with answers. The full set is in the app.

### 1. Sort these statements about projectile motion as True or False.

**Answer:**

- True: Range is maximized at 45° launch angle, Complementary angles (e.g. 30° and 60°) give the same range, The trajectory is a parabola, Horizontal and vertical motions are independent
- False: Time of flight depends on horizontal velocity, Acceleration is zero at the peak

**Why:** Time of flight depends only on vertical velocity; $a = g$ always.

Page: https://tryals.app/practice/physics-i/kinematics-in-two-dimensions-and-projectile-motion/sort-these-statements-about-projectile-motion-as-true-or-false

### 2. Horizontal and vertical motion in projectile flight are treated independently. What makes that separation legitimate?

A. The projectile’s path is a parabola, and parabolas separate by construction
B. Gravity acts only vertically, so the two share only the time
C. The two motions are actually coupled, and the separation is an approximation
D. Air resistance can be neglected, which decouples the components

**Answer:** B. Gravity acts only vertically, so the two share only the time

**Why:** Vector equations separate componentwise when the force has only one component. Time is the sole connection, which is why "how long is it in the air" answers "how far does it go".

Page: https://tryals.app/practice/physics-i/kinematics-in-two-dimensions-and-projectile-motion/horizontal-and-vertical-motion-in-projectile-flight-are-treated

### 3. Complete the following passage by choosing the correct terms from the word bank.

**Answer:** In projectile motion, the horizontal velocity remains **constant** while the vertical acceleration equals **g**.

**Why:** No horizontal force $\to$ constant $v_{x}$; gravity $\to$ $a_y = g$ downward.

Page: https://tryals.app/practice/physics-i/kinematics-in-two-dimensions-and-projectile-motion/complete-the-following-passage-by-choosing-the-correct-terms-from-the