Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf ·...

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Demonstration: Projectile Motion One ball is released from rest at a height h. A second ball is simultaneously fired with a horizontal velocity at the same height. Which ball lands on the ground first? 1. The first ball 2. The second ball 3. Both hit the ground at the same time Neglect air resistance.

Transcript of Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf ·...

Page 1: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Demonstration: Projectile MotionOne ball is released from rest at a height h. A second ball

is simultaneously fired with a horizontal velocity at the same height. Which ball lands on the ground first?

1. The first ball2. The second ball3. Both hit the ground at the same time

Neglect air resistance.

Page 2: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

The independence of x and y

This demonstration shows a powerful idea, that the vertical motion happens completely independently of the horizontal motion.

Page 3: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Equations for 2-D motionAdd an x or y subscript to the usual equations of 1-D motion.

( )

2

2 2

12

2

x ix x

i ix x

x ix x i

v v a t

x x v t a t

v v a x x

= +

= + +

= + − ( )

2

2 2

12

2

y iy y

i iy y

y iy y i

v v a t

y y v t a t

v v a y y

= +

= + +

= + −

Page 4: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Worksheet for today

Let’s see the independence idea in action.

Page 5: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Graphing the vertical motion

Here’s a trick – use the average velocity for a 1-s interval.

Page 6: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Graphing the vertical motion

Here’s a trick – use the average velocity for a 1-s interval.

Page 7: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Graphing the horizontal motion

This one should be a lot easier to do.

Page 8: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Graphing the horizontal motion

The dots are equally spaced, with 1 every 4 boxes.

Page 9: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Graphing the projectile motion

Can you make use of anything we did already?

Page 10: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Graphing the projectile motion

Connect the y-axis dots and the x-axis dots.

Page 11: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Graphing the projectile motion

Connect the dots to get a parabola.

Page 12: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

A ballistics cartWhile a cart is moving horizontally at constant velocity, it fires a ball straight up into the air. Where does the ball land?

1. Behind the cart2. Ahead of the cart3. In the cart

Neglect air resistance.

Page 13: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

The ballistics cart

Again, we see that the vertical motion happens completely independently from the horizontal motion.

Page 14: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Projectile motion

Projectile motion is motion under the influence of gravity alone.

A thrown object is a typical example. Follow the motion from the time just after the object is released until just before it hits the ground.

Air resistance is neglected. The only acceleration is the acceleration due to gravity.

Page 15: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Maximum height

To find the maximum height reached by a projectile, use

Let’s say up is positive, yi = 0, and vy = 0 at maximum height. Also,

Solving for ymax :

The maximum height depends on what planet you’re on, and on the y-component of the initial velocity.

( )= + −2 2 2y iy y iv v a y y

− −= = =

2 2 2 2

max 2 2 2y iy iy iy

y

v v v vy

a g g

.ya g= −

Page 16: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Time to reach maximum height

To find the maximum height reached by a projectile, use

Let’s say up is positive, and vy = 0 at maximum height. Also,

Solving for tmaxheight :

The time to reach maximum height depends on what planet you’re on, and on the y-component of the initial velocity.

0y y yv v a t= +

0 0maxheight

oy y y

y

v v vt

a g g− −

= = =−

.ya g= −

Page 17: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Time for the whole motion

If the projectile starts and ends at the same height, how does the time for the whole trip compare to the time to reach maximum height?

Page 18: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Time for the whole motion

If the projectile starts and ends at the same height, how does the time for the whole trip compare to the time to reach maximum height?

The down part of the trip is a mirror image of the up part of the trip, so:

= =total maxheight2

2 iyvt t

g

Page 19: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Example problem

3.00 seconds after being launched from ground level with an initial speed of 25.0 m/s, an arrow passes just above the top of a tall tree. The base of the tree is 45.0 m from the launch point. Neglect air resistance and assume that the arrow lands at the same level from which it was launched. Use g = 10.0 m/s2.

(a) At what angle, measured from the horizontal, was the arrow launched? Feel free to find the sine, cosine, or tangent of the angle instead of the angle itself if you find

that to be easier.

Page 20: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Step 1

Draw a diagram.

Page 21: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Step 1

Draw a diagram.

Page 22: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Step 2 – Make a data tableKeep the x information separate from the y information.

x-direction y-directionPositive direction ? ?Initial position x0 = ? y0 = ?Initial velocity v0x = ? v0y = ?Acceleration ax = ? ay = ?Displacement and time x = 45.0 m at t

= 3.0 sy = height of tree at t = 3.00 s

Page 23: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Step 2 – Make a data tableKeep the x information separate from the y information.

x-direction y-directionPositive direction right upInitial position x0 = 0 y0 = 0Initial velocity v0x = v0 cosθ v0y = v0 sinθAcceleration ax = 0 ay = -gDisplacement and time x = 45.0 m at t

= 3.00 sy = height of tree at t = 3.00 s

v0 = 25.0 m/s

Page 24: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Part (a) – Find the launch angle.Should we solve the x sub-problem or the y sub-problem?

x-direction y-directionPositive direction right upInitial position x0 = 0 y0 = 0Initial velocity v0x = v0 cosθ v0y = v0 sinθAcceleration ax = 0 ay = -gDisplacement and time x = 45.0 m at

t = 3.00 sy = height of tree at t = 3.00 s

v0 = 25.0 m/s

Page 25: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Part (a) – Find the launch angle.Let’s solve the x sub-problem.

x-directionPositive direction rightInitial position x0 = 0Initial velocity v0x = v0 cosθAcceleration ax = 0Displacement and time x = 45.0 m at

t = 3.00 s

v0 = 25.0 m/s

Page 26: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Part (a) – Find the launch angle.Let’s solve the x sub-problem.

x-directionPositive direction rightInitial position x0 = 0Initial velocity v0x = v0 cosθAcceleration ax = 0Displacement and time x = 45.0 m at

t = 3.00 s

v0 = 25.0 m/s

20 0

12x xx x v t a t= + +

Page 27: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Part (a) – Find the launch angle.Let’s solve the x sub-problem.

x-directionPositive direction rightInitial position x0 = 0Initial velocity v0x = v0 cosθAcceleration ax = 0Displacement and time x = 45.0 m at

t = 3.00 s

v0 = 25.0 m/s

20 0

12x xx x v t a t= + +

0xx v t=

Page 28: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Part (a) – Find the launch angle.Let’s solve the x sub-problem.

x-directionPositive direction rightInitial position x0 = 0Initial velocity v0x = v0 cosθAcceleration ax = 0Displacement and time x = 45.0 m at

t = 3.00 s

v0 = 25.0 m/s

20 0

12x xx x v t a t= + +

045.0 m 15.0 m/s3.00 sx

xvt

= = =

0xx v t=

Page 29: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Part (a) – Find the launch angle.Let’s solve the x sub-problem.

x-directionPositive direction rightInitial position x0 = 0Initial velocity v0x = v0 cosθAcceleration ax = 0Displacement and time x = 45.0 m at

t = 3.00 s

v0 = 25 m/s

20 0

12x xx x v t a t= + +

045.0 m 15.0 m/s3.00 sx

xvt

= = =

0xx v t=0

0

15.0 m/s 3cos25.0 m/s 5

xvv

θ = = =

Page 30: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Back to the data tableLet’s fill in what we know in the table.

x-direction y-directionPositive direction right upInitial position x0 = 0 y0 = 0Initial velocity v0x = +15.0 m/s v0y = v0 sinθAcceleration ax = 0 ay = -gDisplacement and time x = 45.0 m at t

= 3.00 sy = height of tree at t = 3.00 s

v0 = 25.0 m/s

Page 31: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Back to the data tableLet’s fill in what we know in the table.

x-direction y-directionPositive direction right upInitial position x0 = 0 y0 = 0Initial velocity v0x = +15.0 m/s v0y = +20.0 m/sAcceleration ax = 0 ay = -gDisplacement and time x = 45.0 m at t

= 3.00 sy = height of tree at t = 3.00 s

v0 = 25.0 m/s

Page 32: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

How tall is the tree?The arrow barely clears the tree when it passes over the tree 3.00 s after being launched.Find the height of the tree.

1. less than 60.0 m2. 60.0 m3. more than 60.0 m

Page 33: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Part (b) – the height of the treeShould we solve the x or the y sub-problem?

x-direction y-directionPositive direction right upInitial position x0 = 0 y0 = 0Initial velocity v0x = +15.0 m/s v0y = +20.0 m/sAcceleration ax = 0 ay = -gDisplacement and time x = 45.0 m at t

= 3.00 sy = height of tree at t = 3.00 s

v0 = 25.0 m/s

Page 34: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Part (b) – the height of the treeThis a job for the y direction.

y-directionPositive direction upInitial position y0 = 0Initial velocity v0y = +20.0 m/sAcceleration ay = -gDisplacement and time y = height of tree

at t = 3.00 s

Page 35: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Part (b) – the height of the treeThis a job for the y direction.

y-directionPositive direction upInitial position y0 = 0Initial velocity v0y = +20.0 m/sAcceleration ay = -gDisplacement and time y = height of tree

at t = 3.00 s

20 0

12y yy y v t a t= + +

Page 36: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Part (b) – the height of the treeThis a job for the y direction

y-directionPositive direction upInitial position y0 = 0Initial velocity v0y = +20.0 m/sAcceleration ay = -gDisplacement and time y = height of tree

at t = 3.00 s

20 0

12y yy y v t a t= + +

2 210 ( 20.0 m/s) (3.00 s) ( 10.0 m/s ) (3.00)2

y = + + × + − ×

Page 37: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Part (b) – the height of the treeThis a job for the y direction

y-directionPositive direction upInitial position y0 = 0Initial velocity v0y = +20.0 m/sAcceleration ay = -gDisplacement and time y = height of tree

at t = 3.00 s

20 0

12y yy y v t a t= + +

2 210 ( 20.0 m/s) (3.00 s) ( 10.0 m/s ) (3.00)2

y = + + × + − ×

60.0 m 45.0 m 15.0 my = + − = +

Page 38: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Ballistic cart, going downThe ballistic cart is released from rest on an incline, and it shoots the ball out perpendicular to the incline as it rolls down. Where does the ball land?

1. Ahead of the cart2. In the cart3. Behind the cart

Page 39: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Ballistic cart, going upThe ballistic cart is released from rest on an incline, and it shoots the ball out perpendicular to the incline as it rolls down. Where does the ball land?

1. Ahead of the cart (uphill)2. In the cart3. Behind the cart (downhill)

Page 40: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Whole Vectors

Page 41: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Whole Vectors

r = vi t

Page 42: Demonstration: Projectile Motionphysics.bu.edu/~duffy/ns540_fall08_notes05/ns540_session05.pdf · Projectile motion Projectile motion is motion under the influence of gravity alone.

Whole Vectors

r = vi t – 0.5 g t2