Curved Paths

ballistics and circles

§3.3–3.4

Poll Question

Predict: Which ball will land sooner?

A. The ball launched horizontally.

B. The ball dropped vertically.

C. Both will land at the same time.

D. It depends on the launch speed.

Ballistic Trajectories

• Acceleration a = g = 9.8 m/s2 downward– |avertical| = 9.8 m/s2

– |ahorizontal| = 0

• Vertical and horizontal motion are independent and can be treated separately

Ballistic Trajectories

• Flight time t is determined by y and v0y

y = v0y t – 1/2 gt2 (for up = +y)

• Horizontal range is vxt

• Components of velocity vx and vy

For = angle above horizontal of velocity v

vx = v cos(); vy = v sin()

vy

vx

v

Uniform Circular Motion

Speed is constant, direction isn’t

Poll Question

The direction of the acceleration of an object moving at constant speed in a circular path is

A. in the direction of its motion.

B. opposite the direction of its motion.

C. toward the center of its circular path.

D. away from the center of its circular path.

E. a zero vector has no direction.

Poll Question

Two balls with the same speed contact different semicircular guides. Which accelerates the most in the curve?

A. Ball A.

B. Ball B.

C. It’s a tie.

D. Need more information.

A

B

Poll Question

A and B stand the on edge of identical carousels. B has a greater tangential speed. Which accelerates the most?

A. A.

B. B.

C. It’s a tie.

D. Need more information.

A B

B’s greater velocity reverses more often.

Poll Question

Points A and B are different distances from the center of a rotating disk. Which accelerates the most?

A. A.

B. B.

C. It’s a tie.

D. Need more information.

A has a greater velocity change in the same time.

AB

Uniform Circular Motion

• Constant speed in a circular path

• a|| = 0; a = a

• What is the actual vector a?

Uniform Circular Motion

• Define the system

r

Circling Velocity and Acceleration

How far does it move in time dt?

dr

r0

rdt

d

Circling Velocity and Acceleration

v r always

d r0

rdt

vdt

v0

Circling Velocity and Acceleration

Similar triangles

v0

vdt

d

dv

dr

r0

rdt

d

dvv

drr=

Circling Velocity and Acceleration

dvv

drr=

dvdt

= adrdt

= vdvdt

drdt

=1v

1r

=av

vr

a =v2

r

Circling Velocity and Acceleration

Acceleration = a = v2/r

v0

vt

d

dv

Direction of acceleration= direction of velocity + 90°= direction of position+ 180°= opposite direction of r= into the center of the circle!

Poll Question

Which of the following controls are used to accelerate a car? Enter the sum of the correct choices.

32 Steering wheel

16 Power windows

8 Brake pedal

4 Gas pedal

2 Defroster

1 Windshield wipers