Projectile Motion

1. Definition

Projectile motion is the motion of an object that moves under the influence of gravity only after being launched.

Assumptions:

• air resistance is neglected
• gravitational acceleration is constant
• motion occurs in two dimensions

2. Fundamental Idea

Projectile motion combines:

• horizontal motion
• vertical motion

These two motions are independent.

3. Horizontal Motion

In the horizontal direction:

\(a_x=0\)

Therefore horizontal velocity remains constant.

\(v_x=v_{0x}\)​

Horizontal position:

\(x=x_0+v_{0x}t\)

4. Vertical Motion

In the vertical direction:

\(a_y=-g\)

where:

\(g\approx9.8m/s^2\)

Vertical velocity:

\(v_y=v_{0y}-gt\)

Vertical position:

\(y=y_0+v_{0y}-\frac{1}{2}gt^2\)

5. Initial Velocity Components

If an object is launched with speed \(v_0\)​ at angle θ:

\(v_{0x}=v_0\cos\theta\)

\(v_{0y}=v_0\sin\theta\)

These are obtained by resolving the launch velocity into components.

6. Trajectory Shape

Projectile motion produces a parabolic path.

This occurs because:

• horizontal motion is constant velocity
• vertical motion is uniformly accelerated

7. Time of Flight

If launch and landing heights are the same:

\(T=\frac{2v_0\sin\theta}{g}\)​

The projectile spends equal time rising and falling.

8. Maximum Height

At the highest point:

\(v_y=0\)

Using kinematics:

\(H=\frac{{v_0}^2\sin^2\theta}{2g}\)​

9. Horizontal Range

Range is the horizontal distance traveled.

\(R=\frac{{v_0}^2\sin(2\theta)}{g}\)​

Maximum range occurs when:

\(\theta=45^\circ\)

(for equal launch and landing heights)

10. Velocity During Motion

Velocity changes continuously because the vertical component changes.

\(\vec{v}=v_x\hat{i}+v_y\hat{j}\)​

Magnitude of velocity:

\(|\vec{v}|=\sqrt{{v_x}^2+{v_y}^2}\)​​

This follows:

11. Key Physical Insights

Horizontal and vertical motions are independent

Gravity affects only vertical motion.

Acceleration is always downward

Even at the highest point:

• vertical velocity becomes zero
• acceleration remains −g

Symmetry

Without air resistance:

• ascent time = descent time
• speed at equal heights is the same

12. Vector Form

Position vector:

\(\vec{r}(t)=(x_0+v_0t)\hat{i}+\left(y_0+v_0t-\frac{1}{2}gt^2\right)\hat{j}\)​

Velocity vector:

\(\vec{v}(t)=v_{0x}\hat{i}+(v_{0y}-gt)\hat{j}\)​

Summary

Projectile motion is two-dimensional motion under gravity.

Horizontal motion:

\(a_x=0\)

Vertical motion:

\(a_y=-g\)

Key equations:

\(v_{0x}=v_0\cos\theta\)

\(v_{0y}=v_0\sin\theta\)

\(x=v_{0x}t\)

\(y=v_{0y}t-\frac{1}{2}gt^2\)

Projectile motion combines constant horizontal velocity with vertical acceleration, producing a parabolic trajectory.