Components of Velocity

Horizontal Uniformity: In the absence of air resistance, there are no horizontal forces acting on a projectile. Consequently, the horizontal acceleration $a_x$ is zero, and the horizontal velocity $v_x$ remains constant throughout the entire flight.

Vertical Acceleration: Gravity acts as a constant downward force, providing a uniform vertical acceleration $a_y = -g$ (approximately $-9.81$ m/s$^2$). This causes the vertical velocity to decrease linearly as the object rises and increase as it falls.

The Peak Condition: At the maximum height of the trajectory, the vertical component of velocity $v_y$ is momentarily zero. However, the horizontal component $v_x$ remains unchanged, meaning the object is still moving horizontally at the peak.

Step 1: Resolve Initial Velocity

• Calculate the components using $u_x = u \cos \theta$ and $u_y = u \sin \theta$. Ensure the calculator is in degrees or radians as specified.

Step 2: Analyze Vertical Motion

• Use SUVAT equations with $a = -g$ to find the Time of Flight ($t$) or Maximum Height ($H$). For a projectile returning to its launch height, the total time is $T = \frac{2u \sin \theta}{g}$.

Step 3: Analyze Horizontal Motion

• Use the constant velocity formula $d = v \cdot t$ to find the Range ($R$). The range is calculated as $R = u_x \cdot T$, where $T$ is the total time of flight found in the vertical analysis.

Step 4: Link via Time

• Time is the scalar quantity that connects both dimensions. Always solve for time in one dimension first if it is not provided, as it must be identical for both components.

It is vital to distinguish between the properties of the two components to avoid calculation errors:

• Range vs. Height: Range depends on both $u_x$ and the total time $T$, whereas Maximum Height depends only on the initial $u_y$ and gravity.

• The 'Time' Bridge: In almost every multi-step projectile problem, the first goal should be to find the time ($t$). Once you have $t$ from the vertical component, you can immediately use it to find horizontal displacement.

• Symmetry Heuristic: For projectiles that land at the same height they were launched, the time to reach the peak is exactly half of the total time of flight. This can simplify calculations for maximum height significantly.

• Vector Sanity Check: Always ensure that $u_x^2 + u_y^2 = u^2$. If your resolved components do not satisfy the Pythagorean theorem, your trigonometric setup is incorrect.

• Sign Consistency: Choose a direction (usually upwards) as positive. If upwards is positive, then $u_y$ is positive, but acceleration $g$ must be entered as $-9.81$ m/s$^2$.

• Mixing Components: A frequent error is using the total initial velocity $u$ in a horizontal distance formula or a vertical height formula. You must only use $u \cos \theta$ for horizontal and $u \sin \theta$ for vertical calculations.

• Velocity at Peak: Many students assume the total velocity is zero at the peak. In reality, only the vertical component is zero; the object still possesses its full initial horizontal velocity.

• Gravity on Horizontal: Students sometimes mistakenly apply $9.81$ m/s$^2$ to the horizontal component. Gravity is a vertical force and has zero effect on horizontal motion in a vacuum.