Question Question

Position, Velocity & Acceleration Position, Velocity & Acceleration

z Chapter 2 - Kinematics: A description of motion

z Summary of important concepts:

Îposition

Îdisplacement

Îvelocity

» average

» instantaneous

Îacceleration

» average

» instantaneous

Concepts Concepts

z kinematics: A description of motion

z position: your coordinates

z displacement: Δx = change of position

z velocity: rate of change of position

Îaverage : Δx/Δt

Îinstantaneous: slope of x vs. t at a position

z acceleration: rate of change of velocity

Îaverage: Δv/Δt

Îinstantaneous: slope of v vs. t at a position

Question Question

If the average velocity of a car during a trip along a straight road is positive, is it possible for the instantaneous velocity at some time during the trip to be negative?

1 - Yes 2 - No

correct

because the total distance divided by the total time is all that is taken into account when determining the average velocity, you could go forward at a positive velocity, stop, reverse, and continue in the positive direction again and still get a positive average velocity

There are no negative values for instantaneous velocity (speed).

If the velocity of some object is not zero, can its acceleration ever be zero ?

1 - Yes 2 - No

Question Question

correct

Acceleration measures the change in velocity over a given time. If the velocity is not zero, but does not change, acceleration is still zero.

If an object is traveling at a constant speed (such as 40 mph), its acceleration would have to be zero.

Because the formula for the acceleration has the

velocity in the numerator. If it is not zero and time has passed.... there can never be a zero derived.

Is it possible for an object to have a positive velocity at the same time as it has a negative acceleration?

1 - Yes 2 - No

Question Question

correct

It can have a positive velocity and just be slowing down to another lower but still positive velocity

When acceleration and velocity have opposite directions, the object is decelerating

A positive velocity over a change in time can never be negative.

The acceleration is a vector that points in the same direction as the velocity

Concepts & Calculations Concepts & Calculations

z A skydiver is falling straight down, along the negative y direction. During the initial part of the fall, her speed increases from 16 to 28 m/s in 1.5 s.

Which of the following is correct?

1) v>0, a>0 2) v>0, a<0 3) v<0, a>0 4) v<0, a<0

•

following is correct?

1) v>0, a>0 2) v>0, a<0 3) v<0, a>0 4) v<0, a<0

← correct

v a

← correct v a

If speed is increasing, v and a are in same direction.

If speed is decreasing, v and a are in opposite direction.

Not 0 since V_f and V_i are not the same !

= 0

A ball is thrown straight up in the air and returns to its initial position.

During the time the ball is in the air, which of the following statements is true?

1 - Both average acceleration and average velocity are zero.

2 - Average acceleration is zero but average velocity is not zero.

3 - Average velocity is zero but average acceleration is not zero.

4 - Neither average acceleration nor average velocity are zero.

Question Question

correct

V_ave = ΔY/Δt = (Y_f – Y_i) / (t_f – t_i)

a_ave = ΔV/Δt = (V_f – V_i) / (t_f – t_i)

Summary of Concepts Summary of Concepts

z kinematics: A description of motion

z position: your coordinates

z displacement: Δx = change of position

z velocity: rate of change of position

Îaverage : Δx/Δt

Îinstantaneous: slope of x vs. t

z acceleration: rate of change of velocity

Îaverage: Δv/Δt

Îinstantaneous: slope of v vs. t

Constant Acceleration Constant Acceleration

z Textbook sections 2.7 - 2.10

Î1-d motion with constant acceleration

Îfree-fall

Problem

-80 -60 -40 -20 0 20 v (m/s) -300 -200 -100 0 100

0 5 10 15 20

x (meters)

t (seconds)

• Where is velocity zero?

• Where is velocity positive?

• Where is velocity negative?

• Where is speed largest?

• Where is acceleration zero?

• Where is acceleration positive?

position vs. time

velocity vs. time

Concepts & Calculations Concepts & Calculations

z A car is moving along the negative x direction. During part of the trip, the speed increases from 16 to 28 m/s in 1.5 s. Which of the

following is correct?

1) v>0, a>0 2) v>0, a<0 3) v<0, a>0 4) v<0, a<0

•

1) v>0, a>0 2) v>0, a<0 3) v<0, a>0 4) v<0, a<0

If speed is increasing, v and a are in same direction.

If speed is decreasing, v and a are in opposite direction.

← correct

v

a +x

← correct v

a +x

Which of the following statements is most nearly correct?

1 - A car travels around a circular track with constant velocity.

2 - A car travels around a circular track with constant speed.

3- Both statements are equally correct.

Question Question

correct

Speed is just a measure of how fast you are going.

Velocity, however, tells you how fast and in what direction.

In order to have a constant velocity, the car would have to go in the same direction. In this case, a circular track would prevent the driver from going in the same direction.

In a circular track, the velocity will always be equal to zero if a complete circuit is made, so it is constant.

Equations for

Equations for Constant Acceleration Constant Acceleration (text, page 26)

(text, page 26)

z x = x₀ + v₀t + 1/2 at²

z v = v₀ + at

z v² = v₀² + 2a(x-x₀)

z Δx = v₀t + 1/2 at²

z Δv = at

z v² = v₀² + 2a Δx

0 5 10 15 20

0 5 10 15 20

v (m/s)

t (seconds) 0

50 100 150 200

0 5 10 15 20

x (meters)

t (seconds)

0.5 1 1.5 2 a (m/s²)

Lets derive this…

Question Question

An object is dropped from rest. If it falls a distance D in time t then how far will if fall in a time 2t ?

1. D/4 2. D/2 3. D 4. 2D

5. 4D Correct x=1/2 at²

Followup question: If the object has speed V at time t then what is the speed at time 2t ?

1. v/4 2. v/2 3. v 4. 2v 5. 4v

Correct v=at

Demo…

Free- Free -Fall Fall

z constant downward acceleration

z g: acceleration due to gravity

z same for all bodies: g=9.81 m/s²

z a_y = -g = -9.81 m/s²

x y

up

down

Summary of Free-Fall Equations y = y₀ + v_0yt - 1/2 gt²

v_y = v_0y - gt

v_y² = v_0y² - 2gΔy

demo…

Question Question

A ball is thrown vertically upward. At the very top of its trajectory, which of the following statements is true:

1. velocity is zero and acceleration is zero 2. velocity is not zero and acceleration is zero 3. velocity is zero and acceleration is not zero 4. velocity is not zero and acceleration is not zero

correct

The ball is not moving and therefore has a velocity equal to zero. At that instant, the velocity is not changing; resulting in an

acceleration equal to zero.

If acceleration were zero at the top of the trajectory the ball would suddenly hover in mid-air which is not the case.

Since at the top of the trajectory, the ball is not

moving, its velocity would be zero but since acceleration is the measure in the change in velocity, it would not be zero, since the velocity is still changing

Dennis and Carmen are standing on the edge of a cliff.

Dennis throws a basketball vertically upward, and at the same time Carmen throws a basketball vertically downward with the same initial speed. You are standing below the cliff observing this strange behavior. Whose ball is moving fastest when it hits the ground?

1. Dennis' ball 2. Carmen's ball 3. Same

vv₀₀ vv₀₀

Dennis Dennis Carmen

Carmen

HH vv_AA vv_BB

Question Question

Correct: v² = v₀² -2gΔy

Dennis and Carmen are standing on the edge of a cliff.

Dennis throws a basketball vertically upward, and at the same time Carmen throws a basketball vertically downward with the same initial speed. You are standing below the cliff observing this strange behavior. Whose ball hits the ground at the base of the cliff first?

1. Dennis' ball 2. Carmen's ball 3. Same

Question Question

correct

vv₀₀ vv₀₀

Dennis Dennis Carmen

Carmen

y=y₀

vv_AA vv_BB

y=0

v

-gΔt

Looking a little more closely Dennis’ ball….

vv₀₀ DennisDennis

y=y₀

v=v

- gt

Summary

•

equations with constant acceleration

• free-fall

z Δx = v₀t + 1/2 at²

z Δv = at

z v² = v₀² + 2a Δx