Projectile Motion

• Determined by the initial velocity, gravity, and air resistance.

• Footballs, baseballs … any projectile will follow this parabolic trajectory in the x-y plane.

The Independence of x- and y-Motion – Figure 3.9• Notice how the

vertical motion under free fall spaces out exactly as the vertical motion under projectile motion.

• We can treat the x- and y- coordinates separately!

Projectile Motion

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You throw a ball horizontally off a roof. Assuming the ball behaves as an ideal projectile, the time until it lands is determined only by

a) its initial speed and the horizontal distance to the point where it lands.

b) the height of the roof and its initial speed.

c) the height of the roof.

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Clicker question

Where does the apple land?

Driving off a cliff

Vertical , ,

Time for vertical motion

=>

Kicked Soccer Ball Determination of Key Items

Don't forget:

Initial velocity is a 2D vector.

Kicked Soccer Ball

a) = time to reach highest point

Using

b) maximum height

c) Time it takes to reach the ground

d) What is the range

Horizontal range

The path of a particle is symmetric about the highest point

Height and range of a projectilea) Maximum height

Since,

and for

b) Maximum range

So, OR

A cannon can be adjusted to various angle to launch shells at a target 1250m away. What is minimum nozzle velocity? At what angle to launch?

and

A cannon shoots a projectile to a distance of 275m, when it is aimed at 66o. What is the maximum range of the cannon that could achieve with the same projectile and what is the angle requirement?

, and

@ , and So,

Note:

a) (initial y-component of grass hopper’s speed)

(initial speed)

b) Use horizontal motion to find the time for the grass hopper to travel horizontally in air

, and

(Time in air)

Vertical displacement at time

So, *(1.1 s)2 = -4.66 m

(Height of cliff)

A grasshopper leaps into the air from the edge of a vertical cliff

Racing skier

Starting from rest his elevation change is 250 m

How long does it take him to finish the race?Starting from rest his elevation change is 250 m

Vertical acceleration

(down)

Projectile launched from a cliff

What is the velocity at the bottom of the cliff ?Projectile ,

By solving quadratic equation

(time cannot be negative) So,

Horizontal motion

(Does not change)

So,

You and a friend throw two rocks off a bridge. Your friend throws hers with an initial direction 30º below the horizontal. You throw yours with the same initial speed but in a direction 30º above the horizontal. When the two rocks hit the water

a) your friend's is moving faster.

b) yours is moving faster.

c) they are moving at the same speed.

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Clicker question

The monkey jumps from the tree at the moment when the tranquilizer arrow leaves the barrel. Is there a hit?

motions are independent and

Condition:

For a hit the and coordinates must be the same for particular time.

Yes, it is equal

Monkey shoot

Where is the monkey?

a)A

b)B

c)C

d)d

Clicker Question

How far away does the snowball strike the ground?

(Measured horizontally from the edge of the roof)

Let +g be downward ,

And ,

Vertical motion:

Find time in the air

Height

(positive root)

Quadratic equation and

Horizontal motion:

Then,

Helicopter drops parcel

go in the same direction:

Difference

Vertical motion

(time to hit the ground)

Horizontal motion

(release distance is 55.4m behind the car)

Parcel is always below the helicopter towards the ground which is positive displacement and the angle of hit is:

Parcel delivered by helicopter

Center-seeking Acceleration

OR

The magnitude of the average acceleration changes within

Instantaneous acceleration: (Drop index)

(Radial acceleration)

Circular Motion

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The pilot takes a vertical loop with a velocity of 700 km/hr. What radius R should the plane fly and not exceed acceleration ?

(6g is the acceleration where most persons black out)

(radial acceleration)

A Problem to Try on Your Next Vacation –Example 3.9

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• Uniform circular motion applied to a daring carnival ride.

¿(7.9

𝑚𝑠 )

2

5.0𝑚

¿2𝜋 (5𝑚 )

4.0 𝑠=7.9

𝑚𝑠

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crossing a stream

An Airplane in a Crosswind – Example 3.10

• A solved application of relative motion.

• This is just velocity vector addition.

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