4.3 - Projectile Motion Launched at an Angle

There are 2 general types of projectile motion situations:

1. Object launched horizontally















2. Object launched at an angle

Since a projectile moves in 2-dimensions, it therefore has 2 components just like a resultant vector:


Horizontal


Vertical

With no gravity the projectile would follow the straight-line path
(dashed line).

But because of gravity it falls beneath this line the same vertical
distance it would fall if it were released from rest.

Velocity
The velocity of a projectile is shown at various points along its path. Notice that the vertical component changes while the horizontal component does not. Air resistance is neglected.

Height
For the component vectors of the angled projectile motion, the
horizontal component is always the same and only the vertical
component changes.

Height
At the top of the path the vertical component shrinks to zero.

Together, these components produce
what is called a trajectory or path. This
path is parabolic in nature.

Component

Magnitude

Direction

Horizontal

Constant

Constant

Vertical

Changes

Changes

Range
The angle at which the projectile is launched affects the
distance that it travels.

Horizontal Ranges
Projectiles that are launched at the same speed but at
different angles reach different heights (altitude) above the
ground.

They also travel different horizontal distances, that is, they
have different horizontal ranges.

Horizontal Range
Both projectiles have the same launching speed.
The initial velocity vector has a greater vertical component than
when the projection angle is less.

•

This greater component results in a higher path.

•

The horizontal component is less, so the range is less.

The paths of projectiles launched at the same speed but at different
angles. The paths neglect air resistance.

The same range is obtained for two different projection angles—
angles that add up to 90°.

An object thrown into the air at an angle of 60° will have the same
range as at 30° with the same speed.

Max Range
Maximum range is attained at an angle of 45°

Max Height
Without air resistance, a projectile will reach maximum height in
the same time it takes to fall from that height to the ground.

Acceleration
The deceleration due to gravity going up is the same as the
acceleration due to gravity coming down.

Speed
The projectile hits the ground with the same speed it had when it
was projected upward from the ground provided it lands with the
a displacement of zero. (Lands at the same distance from the
ground that it took off from)

Thinker!
A projectile is launched at an angle into the air. Neglecting air
resistance, what is its vertical acceleration? Its horizontal
acceleration?

Hotspot

At what point in its path does a projectile have minimum speed?

Vx is constant

Vy decreases
on the way
upward

Vy increases on the
way down,

Vy =0 at the

Component

Magnitude

Direction

Horizontal

Constant

Constant

Vertical

•Decreases up

•0 @ top

•Increases down

Changes

Thinker!
At what point in its path does a projectile have minimum speed?

Answer
Top of the parabolic path

Open Ended

A projectile is launched at an angle into the air. Neglecting air
resistance, what is its vertical acceleration? Its horizontal
acceleration?

Projectiles launched at angles summary:

•The horizontal velocity is constant.

•It rises and falls in equal time intervals.

•It reaches maximum height in half the total time.

•Gravity only affects the vertical motion.

•If it begins and ends at ground level, the “dy”displacement is ZERO: dy = 0

Components
Since the projectile was launched at an angle, the velocity
MUST be broken into components!!!

vi

vix

viy

θ

Vix = Vi cos θ

Viy = Vi sin θ

Formulas
You will still use the kinematic equations, but YOU MUST use
COMPONENTS in the equation.

tV

d
ix

x



cos
i

ix
V

V



2

2
1

ta

tV

d
y

iy

y




sin
i

iy
V

V 

vi

vix

viy

θ

YOU WILL NEVER USE ViIN YOUR KINEMATIC EQUATIONS!!!!!!!!!!!!!

What is the first thing you need to do if a projectile is
launched at an angle?

You must break the initial velocity (Vi) into components:

Vix and Viy

Open Ended

What is the first thing you need to do if a projectile is
launched at an angle?

Question #1
A place kicker kicks a football with a speed of 20 m/s and at an angle
of 53°.

a.

What are the horizontal and vertical components of the initial
speed?

b.

How long is the ball in the air?

c.

How far away does it land?

d.

How high does it travel?

​1. Horizontal and Vertical Components of the Initial Speed Given:

• Initial speed (v_0​) = 20 m/s

• Angle (θ) = 53°

Horizontal Component (v_0x​): v_0x=v₀⋅cos⁡(θ)
=20⋅cos⁡(53°)≈20⋅0.6018 ≈ 12.04 m/s

Vertical Component (v_0y​): v_0y=v₀​⋅sin(θ)
=20⋅sin⁡(53°)=20⋅sin(53°)≈20⋅0.7986≈ 15.97 m/s

2. Time in the Air

To find the time the ball is in the air, we use the vertical motion. The ball will go up and come back down, so we can use the formula for the time to reach the maximum height and then double it (since the time up equals the time down).

Using the formula: t=v_0y/g where g=9.8 m/s² (acceleration due to gravity).

Time to reach maximum height: t_up=15.97/9.8 ≈1.63 s

Total time in the air: t_total=2×t_up≈2×1.63 ≈ 3.26 s

3. Horizontal Distance (Range)

The horizontal distance can be found using the horizontal component of the velocity and the total time in the air.

=12.04×3.26≈39.24 m

4. Maximum Height

The maximum height can be found using the vertical component of the velocity.

Using the formula: h=v²_0y/2g​​
h=15.97²/2×9.8 ≈ 255.04/19.6 ≈ 13.01 m

Example #2
A body is projected upward from the level ground at an angle of 50° with the
horizontal has an initial speed of 40 m/s.

a.

What are the horizontal and vertical components of the initial speed?

b.

How long will it be before it hits the ground?

c.

How far from the starting point will the object hit the ground?

d.

What is the maximum height it reached in the air?

a.

Vix = 25.71 m/s; Viy = 30.64 m/s

b.

6.25 s

c.

dx = 160.62 m

d.

dymax = 47.85 m

Multiple Choice

$$V_{ox}\ =\ ?$$  

• $$V_o\cos\theta$$  

• $$V_o\sin\theta$$  

Multiple Choice

$$V_{oy}\ =\ ?$$  

• $$V_o\cos\theta$$  

• $$V_o\sin\theta$$  

Multiple Choice

Why is g negative?

$$g\ =\ -9.8\ \frac{m}{s^2}$$  

• It has no effect on the motion of the object.

• It has a downward direction.

Multiple Choice

During the upward flight, velocity _____.

• increases

• decreases

Multiple Choice

During the downward flight, velocity _____.

• increases

• decreases

Multiple Choice

What is vertical velocity of the object at the peak or highest vertical distance?

• $$V_y\ =\ 0$$  

• $$V_y\ =\ 100$$  

Multiple Choice

height

• $$d_x$$  

• $$d_y$$  

Multiple Choice

The more mass an object has, the faster it will fall.

• True
• False

Multiple Choice

If you throw a baseball straight up, what is its velocity at the highest point?

• about 10 m/s²

• about 10 m/s

• 0 m/s²

• 0 m/s

Multiple Choice

If you throw a baseball straight up, what is its acceleration at the highest point?

• about 10 m/s²

• about 10 m/s

• 0 m/s²

• 0 m/s

Multiple Choice

A frog jumps vertically straight up. Which of the following is true?

• gravity gives the frog a vertical speed of -10 m/s

• gravity gives the frog a vertical acceleration of -10 m/s²

• the frog has a horizontal speed of -10 m/s

• the frog has a horizontal acceleration of -10 m/s²