Projectiles: Two Independent Motions | CAIE 9702 Paper 2 Notes

Paper 2

Physics9702

Projectiles: Two Independent Motions

A is an object thrown or launched and then left to move under gravity alone. This model starts after release and ignores air resistance.

Its motion splits into two separate problems: constant velocity across, constant acceleration down. The only quantity the two share is the time.

Why the two directions are separate

Gravity pulls straight down. There is no horizontal force (air resistance is ignored), so the horizontal velocity never changes.
The vertical motion is ordinary free fall: acceleration $g$ downward, whatever the horizontal motion is doing.
At the highest point, only the vertical velocity is zero. The horizontal velocity remains, so the total velocity is not zero, and the acceleration is still $g$ downward.

Horizontal velocity stays constant for the whole flight (violet). Vertical velocity decreases to zero at the top, then increases downward. Acceleration remains downward throughout.

The test of this idea: drop one ball and throw another sideways from the same height at the same instant. They land together. The sideways throw does not change the fall.

The sideways throw does not change the fall. At every instant both balls are at the same height (dashed links), so they hit the floor at the same time.

Where the independence comes from

Newton's second law works direction by direction. The only force is gravity, which acts straight down, so only the vertical velocity changes. The horizontal direction has no force, so no acceleration, so constant velocity for the whole flight.

Splitting the launch velocity

A launch at angle $\theta$ is resolved once, at the start, using the components from lesson 1.8:

u_x = u\cos\theta

Symbols

$u$ = launch speed (m s⁻¹)
$\theta$ = launch angle above the horizontal (degrees)
$u_x$ = horizontal component (stays constant) (m s⁻¹)

u_y = u\sin\theta

Symbols

$u$ = launch speed (m s⁻¹)
$\theta$ = launch angle above the horizontal (degrees)
$u_y$ = initial vertical component (changes under g) (m s⁻¹)

Split the initial velocity once: u cos θ horizontally and u sin θ vertically. The horizontal component remains constant; gravity changes the vertical component.

Drag the launch speed and angle below to see the two motions run at the same time:

Projectile Explorer

This model ignores air resistance and lands at the launch height. During one flight, the horizontal component stays constant while gravity changes only the vertical component.

Launch speed u20 m s⁻¹

Launch angle θ45°

uₓ = u cos θ

14.1 m s⁻¹

uᵧ = u sin θ

14.1 m s⁻¹

time of flight

2.88 s

range R

40.8 m

Your turnAdapted from 9702/11/M/J/24 Q7 · 1 mark

A golf ball flies through the air; air resistance is ignored. State what happens to the horizontal and vertical components of its velocity. (9702/11/M/J/24 Q7)

Show worked answer

Answer

The horizontal component remains constant. The vertical component changes at a constant rate because the ball has constant downward acceleration.

Solve every projectile as two separate motions joined by one shared time: constant velocity across, free fall down.

Projectiles: Two Independent Motions

A is an object thrown or launched and then left to move under gravity alone. This model starts after release and ignores air resistance.

Its motion splits into two separate problems: constant velocity across, constant acceleration down. The only quantity the two share is the time.

Why the two directions are separate

Gravity pulls straight down. There is no horizontal force (air resistance is ignored), so the horizontal velocity never changes.
The vertical motion is ordinary free fall: acceleration $g$ downward, whatever the horizontal motion is doing.
At the highest point, only the vertical velocity is zero. The horizontal velocity remains, so the total velocity is not zero, and the acceleration is still $g$ downward.

Horizontal velocity stays constant for the whole flight (violet). Vertical velocity decreases to zero at the top, then increases downward. Acceleration remains downward throughout.

The test of this idea: drop one ball and throw another sideways from the same height at the same instant. They land together. The sideways throw does not change the fall.

The sideways throw does not change the fall. At every instant both balls are at the same height (dashed links), so they hit the floor at the same time.

Where the independence comes from

Splitting the launch velocity

A launch at angle $\theta$ is resolved once, at the start, using the components from lesson 1.8:

u_x = u\cos\theta

Symbols

$u$ = launch speed (m s⁻¹)
$\theta$ = launch angle above the horizontal (degrees)
$u_x$ = horizontal component (stays constant) (m s⁻¹)

u_y = u\sin\theta

Symbols

$u$ = launch speed (m s⁻¹)
$\theta$ = launch angle above the horizontal (degrees)
$u_y$ = initial vertical component (changes under g) (m s⁻¹)

Split the initial velocity once: u cos θ horizontally and u sin θ vertically. The horizontal component remains constant; gravity changes the vertical component.

Drag the launch speed and angle below to see the two motions run at the same time:

Projectile Explorer

This model ignores air resistance and lands at the launch height. During one flight, the horizontal component stays constant while gravity changes only the vertical component.

Launch speed u20 m s⁻¹

Launch angle θ45°

uₓ = u cos θ

14.1 m s⁻¹

uᵧ = u sin θ

14.1 m s⁻¹

time of flight

2.88 s

range R

40.8 m

Your turnAdapted from 9702/11/M/J/24 Q7 · 1 mark

A golf ball flies through the air; air resistance is ignored. State what happens to the horizontal and vertical components of its velocity. (9702/11/M/J/24 Q7)

Show worked answer

Answer

The horizontal component remains constant. The vertical component changes at a constant rate because the ball has constant downward acceleration.

Solve every projectile as two separate motions joined by one shared time: constant velocity across, free fall down.