PhysicsInstructor: Dr. Tatiana Erukhimova

1D, 2D Motion

Motion in One Dimension

We consider a particle

)(tx - as time goes, the position of the particle changes

Velocity is the rate at which the position changes with time

Average velocity:

t

x

tt

txtxv

initialfinal

initialfinalave

)()(

dt

tdxv

)(

Acceleration is the rate at which the velocity changes with time

Average acceleration

t

v

tt

tvtva

initialfinal

initialfinalave

)()(

dt

tdva

)(

)0()0(2

1)( 2 xtvtatx c

)0()( vtatv c

))()((2)()( 1212

22 txtxatvtv c

A ball is dropped vertically down (no air resistance)

from height H. Find the position x(t) and velocity v(t) of the ball as a function of time. How long the ball is in the air?

A person throws a ball upward into the air with an initial velocity of 15 m/s. Calculate

a) How much time it takes for the ball to reach the maximum height?

b)How high does it go?c) How long is the ball in the air before it comes back

to handsd)The velocity of the ball when it returns to the

thrower’s hande) At what time t the ball passes a point 8.00 m above

the person’s hand

Suppose that a ball is dropped from a tower 70.0 m high. How far will it have fallen after 2s?

What is its velocity at 2 s?

Suppose the ball is thrown downward with an initial velocity of 3 m/s instead of being dropped. Find

1) Position after 2s2) Speed after 2s

An experimental vehicle starts from rest (v0=0) at t=0 and accelerates at a rate given by a=7t, where t is time.

What is1) Its velocity at any time moment2) Its position 2s later.

A particle moves along the x axis. Its position as a function of time is x = 6t+8.5t2

where t is in seconds and x in meters. What is the acceleration as a function of time?

A cannon at the origin points up at an angle θ with the x axis. A shell is fired which leaves the barrel with a velocity of magnitude Vm.

a) When does the shell reach its maximum height?b)What is the maximum height?c) What is the range (horizontal distance)?d)What is the velocity of the shell when it hits the

ground?

A rock is thrown from the roof of a building with a velocity V0 at an angle from the horizontal. The building has height H. You can ignore air resistance. Calculate the magnitude of the velocity of the rock just before it strikes the ground, and show that this speed is independent of .

A physics professor did daredevil stunts in his spare time. His last stunt was an attempt to jump across a river on a motorcycle. The takeoff ramp was inclined at 53.00, the river was 40.0 m wide, and the far bank was 15.0 m lower than the top of the ramp. The river itself was 100 m below the ramp. You can ignore air resistance. a) What should his speed have been at the top of the ramp to have just made it to the edge of the far bank? b) If his speed was only half the value found in (a), where did he land?

A can drops from a magnet just when a bullet is shot from a gun: Find the angle that the gun must be aimed at to hit the can.

An object’s acceleration is given by

jtita 2

If the object starts at t=0 at the origin with an initial velocity of magnitude vi directed at θ above the x axis, what are its velocity and position at any time?

Problem 3 from the Pre-Exam: A robot has been programmed so that it has an acceleration given by

in the coordinate system below. Here is a known constant. The robot is to be placed at the point x = 0, y = A at t = 0. Here A is a known constant.

a) What is the robot’s position as a function of time if at t = 0 the robot was given some initial velocity ?

(C, D)

x

y

1V

1V

jtia 0

b) What must be the magnitude of the robot’s initial velocity, v1 , if it leaves the initial point at the known angle θ as shown and can pick up some object at the point x = C, y = D? (Here C, and D are known constants).

1V

A block is placed at rest on an inclined plane of height H with angle Θ with horizontal. If the surface is frictionless, how fast will the block be moving when it gets to the bottom of the plane?

€

H

How fast will the block be moving when it gets to the bottom of the plane if the coefficient of friction between the block and the plane is μ?

A block of mass m is given an initial velocity V0 up an inclined plane with angle of incline Θ. If there is a non-zero coefficient of friction between the plane and the block, μ, how long does it take for the block to return to its original position?

Falling with air resistance

€

a =dv

dt= g − kv 2

Terminal Velocity with Coffee Filters

€

mg − Fr = ma

where is the resistance force.

€

Fr

€

a = g −Fr

m

1. A penny and a quarter dropped from a ladder land at the same time (air resistance is negligible).

2. A coin dropped in a coffee filter from a ladder lands later than a coin without coffee filter (the terminal velocity is smaller for larger cross-section area).

3. A quarter dropped in a coffee filter will land faster than a penny in a coffee filter (the terminal velocity is larger for larger mass)

4. Two identical coins dropped in coffee filters of different diameters land at different times (the terminal velocity is smaller for larger cross-section area).

Resistance force:

€

Fr = γAv 2

A – area of the projectile

€

γ=0.25 N × s2 /m4For a spherical projectile in air at STP:

Terminal velocity:

€

a = g −Fr

m= 0

Fr = mg

γAv 2 = mg

€

vT =mg

γA

A 70-kg man with a parachute: vT ~ 5 m/sA 70-kg man without a parachute: vT ~ 70 m/s