3.4 Projectile Motion

Projectile Motion

• A projectile is anything launched, shot or

thrown---i.e. not self-propelled. Examples:

a golf ball as it flies through the air, a

kicked soccer ball, a thrown football, a

speeding bullet.

• The path that the projectile follows is

called a trajectory.

2-D Projectile Motion: Basic Equations

Assumptions:

• ignore air resistance

• g = 9.80 m/s2, downward

• ignore Earth’s rotation

If y-axis points upward, acceleration in

x-direction is zero and acceleration in

y-direction is -9.80 m/s2

2-D Projectile Motion

𝑥𝑓 = 𝑥𝑖 + 𝑣𝑥𝑖𝑡

𝑦𝑓 = 𝑦𝑖 + 𝑣𝑦𝑖𝑡 −1

2𝑔𝑡2

𝑣𝑥𝑓 = 𝑣𝑥𝑖

𝑣𝑦𝑓 = 𝑣𝑦𝑖 − 𝑔𝑡

Where g = 9.80 m/s2.

This motion is for a passive projectile. A

missile or vehicle (e.g. airplane) may also

have self-controlled acceleration (ax, ay, az).

2-D Projectile Motion

𝑥 = 𝑥𝑖 + 𝑣𝑥𝑡

𝑦 = 𝑦𝑖 + 𝑣𝑦𝑖𝑡 −1

2𝑔𝑡2

Motion in the x-y plane will be parabolic,

because horizontal position is linear with

time, and vertical position is quadratic.

𝑡 = 𝑥 − 𝑥𝑖 𝑣𝑥

𝑦 = 𝑦𝑖 + 𝑣𝑦𝑖 𝑥 − 𝑥𝑖 𝑣𝑥 −1

2𝑔 𝑥 − 𝑥𝑖 𝑣𝑥

2

𝑦 = 𝐶 + 𝐵𝑥 + 𝐴𝑥2

A dropped object and

an object launched

horizontally will fall at

the same rate. Motion

in the horizontal and

vertical directions are

independent.

The vertical motion

dictates the “time of

flight”.

2-D Projectile Motion

Demonstration!

A dropped object keeps up with you

(neglecting air resistance)

A new extreme sport?

(a) Write the horizontal and vertical equations of

motion for v0 = 1.30 m/s. (b) What is the x and y

position of the basketball at t = 0.500 s?

Two friends jump at the same time.

(a) Who hits the water first? (b) Who

hits the water with greater speed?

Launch Angle

Launch angle: direction of initial velocity with

respect to horizontal. This affects the

location of the vertex.

2-D Projectile Motion

If an object is launched at an initial angle of θ0

with the horizontal, the analysis is similar except

that the initial velocity has a vertical component.

Launch Angle

For angle θ, v0x = v0 cos θ and v0z = v0 sin θ.

This gives the equations of motion:

𝑥𝑓 = 𝑥𝑖 + 𝑣0 cos 𝜃 𝑡

𝑦𝑓 = 𝑦𝑖 + 𝑣0 sin 𝜃 𝑡 −1

2𝑔𝑡2

𝑣𝑥 = 𝑣0 cos 𝜃 = constant

𝑣𝑦𝑓 = 𝑣0 sin 𝜃 − 𝑔𝑡

Range

The range of a projectile is the horizontal

distance it travels before landing:

Range = 𝑥𝑓 − 𝑥𝑖 = 𝑣0 cos 𝜃 𝑡

𝑦𝑓 = 𝑦𝑖 + 𝑣0 sin 𝜃 𝑡 −1

2𝑔𝑡2

Solve the y- equation for t, defining ∆𝑦 = 𝑦𝑓 − 𝑦𝑖

𝑡 =𝑣0 sin 𝜃 ± 𝑣0 sin 𝜃

2 − 2𝑔∆𝑦

𝑔

Range = 𝑣0 cos 𝜃𝑣0 sin 𝜃± 𝑣0 sin 𝜃 2−2𝑔∆𝑦

𝑔

Let’s consider special cases 𝜃 = 0 and ∆𝑦 = 0 .

Range: Special Case 𝜃 = 0

Range = 𝑣00+ 0−2𝑔∆𝑦

𝑔

= 𝑣02ℎ

𝑔

So to jump a chasm

you need

𝑣0 ≥ 𝑤𝑔

2ℎ

Range: Special Case ∆𝑦 = 0

Range = 𝑣0 cos 𝜃𝑣0 sin 𝜃+ 𝑣0 sin 𝜃 2+0

𝑔=

𝑣02 2 cos 𝜃 sin 𝜃

𝑔=𝑣0

2 sin 2𝜃

𝑔

Maximum range occurs

when

sin 2𝜃 = 1i.e.

𝜃 = 45°then

𝑅 = 𝑣02 𝑔

Range: Special Case ∆𝒚 = 𝟎:

The range is a maximum when θ = 45°:

Reality Check

Air resistance causes horizontal and vertical

velocities to decelerate.

Wind also affects the range of a projectile.

Spin on a projectile also affects its motion

(“lift” due to relative velocity + Bernoulli’s

Principle)

Projectiles get their initial

velocity from…

• Elasticity – thrown objects, bow, slingshot,

torsion catapult, toy spring dart gun

• Rotational motion - sling

• Gravity – trebuchet, ramp (e.g. ski jump)

• Expanding gas – blow gun, cannon, rifle

• Electromagnetic force – rail gun, coil gun

Self-powered rockets and guided missiles

are not projectiles.

Hitting a Target

If the rifle is fired directly at the target in a horizontal direction, will the bullet hit the center of the target?

Does the bullet fall during its flight?

Which of the two trajectories shown will result in a longer time for the ball to reach home plate?

a) The higher trajectory.

b) The lower trajectory.

c) They will take the same time.

a) The higher trajectory takes

longer. The time of flight is

determined by the initial vertical

velocity component which also

determines the maximum height

reached.

ConcepTest 4.2 Dropping a Package

You drop a package from

a plane flying at constant

speed in a straight line.

Without air resistance, the

package will:

1) quickly lag behind the plane

while falling

2) remain vertically under the

plane while falling

3) move ahead of the plane while

falling

4) not fall at all

You drop a package from

a plane flying at constant

speed in a straight line.

Without air resistance, the

package will:

1) quickly lag behind the plane

while falling

2) remain vertically under the

plane while falling

3) move ahead of the plane while

falling

4) not fall at all

Both the plane and the package have

the same horizontal velocity at the

moment of release. They will maintain

this velocity in the x-direction, so they

stay aligned.

Follow-up: What would happen if air resistance were present?

ConcepTest 4.2 Dropping a Package

• 3.4 Clicker Questions