1.4 Projectile Motion and Fluid Resistance

Model ideal projectile motion with components, then contrast it with drag-influenced motion and terminal speed.

Estimated time: 26 minutes

Horizontal and Vertical Motions Are Independent

In the ideal model (neglecting air resistance), projectile motion is two simultaneous one-dimensional motions sharing the same time variable. Horizontally there is no acceleration, so horizontal velocity is constant. Vertically there is constant downward acceleration g.

This is why a ball dropped and a ball launched horizontally from the same height hit the ground at the same time in the ideal model: their vertical motions are identical.

Component Equations for Velocity and Position

$v_x=u\cos\theta,\quad v_y=u\sin\theta-gt,\quad x=ut\cos\theta,\quad y=ut\sin\theta-\frac{1}{2}gt^2$

Choose the launch point as origin whenever possible to simplify interpretation.

At the highest point, vertical velocity is zero but horizontal velocity is still nonzero, so the projectile is still moving. Time to peak, maximum height, and range can all be derived quickly from these component equations instead of memorized in isolation.

What Air Resistance Changes

Real projectiles experience drag opposite their velocity. A common simplified model uses drag proportional to speed at lower speeds, giving a resistance force F = kv. As speed rises, drag rises, reducing both peak height and range compared with the ideal parabola.

$mg = kv_T$

When drag equals weight during vertical fall, acceleration becomes zero and speed settles to terminal speed v_T.

With drag, the path is no longer symmetric and landing angle is steeper. That mismatch between ideal and real trajectories is not a failure of kinematics; it is a signal that force models have become richer.

Simulation: Projectile Components and Flight Outcomes

Change launch angle and speed, then compare range, peak height, and time of flight in an idealized model.

Projectile Lab

Speed u: 20.0 m/sAngle θ: 35.0°Gravity g: 9.81 m/s²Initial height: 0.0 m

uₓ

16.38 m/s

uᵧ

11.47 m/s

Flight time

2.34 s

Range

38.32 m

Max height

6.71 m

Trajectory

Test Yourself

A projectile is launched horizontally from 20 m with speed 15 m/s. Using g = 10 m/s^2, enter the horizontal distance before impact in meters.

Hint: Find time from vertical motion first, then use x = v_x t.

Your numerical answer (m)

1.3 Graphs of Motion

Chapter 1 Wrap-Up