Projectile Motion

• Projectile motion: a combination of horizontal motion with constant horizontal velocity and vertical motion with a constant downward acceleration due to gravity.

• Projectile motion refers to the motion of an object that is thrown, or projected, into the air.

• We restrict ourselves to objects thrown near the Earth’s surface so that gravity can be considered to be constant.

Projectile Motion

• The motion of a projectile is determined only by the object’s initial velocity and gravity.

• The vertical motion of a projected object is independent of its horizontal motion.

• The vertical motion of a projectile is nothing more than free fall.

• The one common variable between the horizontal and vertical motions is time.

Horizontal velocity vector (remains constant)

Vertical velocity vector (steadily increasing)

Projectile motion problems are best solved bytreating horizontal and vertical motion separately.

*IMPORTANT*Gravity only affects vertical motion.

There are two general typesof projectile motion situations.

1. object launched horizontally2. object launched at an angle

Path of a Projectile

• A projectile moves horizontally with constant velocity while being accelerated vertically. A right angle exists between the direction of the horizontal and vertical motion; the resultant motion in these two dimensions is a curved path. – The path of a projectile is called its trajectory.– The trajectory of a projectile in free fall is a

parabola.

Object Launched Horizontallyvx = initial horizontal velocity

Rx = horizontal range

h = initial height above ground

t = total time in the airIMPORTANT FACTS

There is no horizontal acceleration.There is no initial vertical velocity.The horizontal velocity is constant.

Time is the same for both vertical and horizontal.

horizontalRx = vxt

vertical

h = 0.5gt2

Motion in 2D…

• Throw a ball on a moving truck demo• Monkey shooting

For Objects Shot Horizontally:

• vx constant

• Dy negative Dy = -height

tvxsm

0vtx

v

x

yix

2tg0.5Δy

For Objects Shot Horizontally:

• When hits at bottom:– Vyf should be negative– vo = resultant velocity

2yf

2xo

yf

vvv

gtv

For Objects Shot Horizontally:

• q with horizontal:

• q with vertical:

x

yf1

v

vtan

yf

x1

vv

tan

Projectile Motion

Motion in two dimensions

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When a projectile is launched at an angle the object moves both up and down and side to side at the same time

The vertical component (Vy) controls how high the object rises. This is the objects maximum height. It occurs half way through the projectiles path.

**The path of a projectile is called its trajectory

This means, when a projectile is launched at an angle, it’s velocity has a vertical and horizontal component

Since the maximum height (dy) occurs half way through the trajectory, vertical equations use time for only half the trip.

total time=4s

ty=2s

The horizontal component (Vh) controls how far over the object travels. This is called the object’s range. It occurs at

the end of the entire trajectory.

Since the object’s range (dx), occurs over the entire trip, horizontal equations must use the time for the total trip.

tx=4s

Changing the angle at which the projectile is launched effects the vertical and horizontal distances the object travels

Fill in the x & y velocity vectors

Path of a Projectile

Path of a Projectile

• vo = initial velocity or resultant velocity• vx = horizontal velocity• vyi = initial vertical velocity• vyf = final vertical velocity• R= maximum horizontal distance (range)• x = horizontal distance• Dy = change in vertical position• yi = initial vertical position• yf = final vertical position• q = angle of projection (launch angle)• H = maximum height• ag = gravity = 9.8 m/s2

Path of a Projectile

• The horizontal distance traveled by a projectile is determined by the horizontal velocity and the time the projectile remains in the air. The time the projectile remains in the air is dependent upon gravity.

• Immediately after release of the projectile, the force of gravity begins to accelerate the projectile vertically towards the Earth’s center of gravity.

Path of a Projectile

• The velocity vector vo changes with time in both magnitude and direction. This change is the result of acceleration in the negative y direction (due to gravity). The horizontal component (x component) of the velocity vo remains constant over time because there is no acceleration along the horizontal direction

• The vertical component (vy) of the velocity vo is zero at the peak of the trajectory. However, there is a horizontal component of velocity, vx,

at the peak of the trajectory.

Path of a Projectile

Horizontal velocity component:

• vx is constant because there is no acceleration in the horizontal direction if air resistance is ignored.

cos ox vv

Vertical velocity component:

• At the time of launch:

• After the launch:

– If vy positive, direction of vertical motion is up; if vy negative, direction of vertical motion is down; if vy = 0, projectile is at highest point.

θsinvv oyi

tasinθvv goyf

Horizontal position component:

• If you launch the projectile horizontally:– then vo = vx

– vyi = 0 m/s– = 0o

tθcosvx

tvx

o

x

Vertical position component:

2

goif

2gyiif

if

ta0.5tθsinvyy

ta0.5tvyy

yyΔy

Determining vo from vx and vy

• If the vertical and horizontal components of the velocity are known, then the magnitude and direction of the resultant velocity can be determined.

• Magnitude: 2

y2

xo vvv

Determining vo from vx and vy

• Direction: from the horizontal

• Direction: from the vertical

x

y1

v

vtanθ

y

x1

v

vtanθ

Range and Angle of Projection

Range and Angle of Projection

• The range is a maximum at 45 because sin (2·45) = 1.

• For any angle other than 45, a point having coordinates (x,0) can be reached by using either one of two complimentary angles for , such as 15 and 75 or 30 and 50 .

Range and Angle of Projection

• The maximum height and time of flight differ for the two trajectories having the same coordinates (x, 0).

• A launch angle of 90° (straight up) will result in the maximum height any projectile can reach.

For Situations In Which Dy Positive

• At any point in the flight: sinvvcosvv oyiox

2yf

2xo

x

x

gyiyf

22o

g2

2gyi

vvv

constantv

tvx

tavv

cosθv2

axtanθxΔy

ta0.5tvΔy

What angle gives the projectile maximum range?

Excel Spreadsheet

Angle Time Velocity Vertical Distance Range15 2 4.5 2.33 1.0330 2 4.5 4.50 1.7945 2 4.5 6.36 2.0760 2 4.5 7.79 1.7975 2 4.5 8.69 1.03

And the winner is ….

• A 45 degree angle gives the projectile maximum range

• In nature, frogs naturally jump at approximately a 45 degree angle to help them cover maximum distance on flat ground