Part 1: Introduction & 1-D Kinematics



Part 1: Introduction & 1-D Kinematics

1.1: Dimensional Analysis/Units

Conversions

Multiply by conversion factors so unwanted units cancel.

Example:

Light travels in free space at a speed of c = 3x108 meters per second. Convert this speed to kilometers per year and miles per second.

[pic]

[pic]

1.2 Vectors

Scalar physical quantities have just a size with appropriate units.

Vector physical quantities have a size (magnitude) and a direction. The magnitude has the units of the quantity. The direction is given by one angle (2-D) or two angles (3-D), usually in degrees.

Position is a vector physical quantity. In 2-D, we can represent as follows.

Polar Form: [pic]

Component Form: [pic]

Converting from Polar to Component Form

[pic]

[pic]

Converting from Component to Polar Form

[pic]

[pic] NOTE: If the x-component is negative, add 180( to get correct angle.

Adding Vectors

Graphically

Place vectors tail-to-tip. Resultant vector goes from tail of first vector to tip of last vector in sum.

Analytically

1. The vectors must be in component form. Convert to polar form if necessary.

2. Add x-components to get the x-component of the resultant vector.

3. Add y-components to get the y-component of the resultant vector.

4. If you want the resultant in polar form, convert from component to polar form.

Subtracting Vectors

Graphically

Reverse the direction of the subtracted vector. Now add as usual.

Analytically

1. The vectors must be in component form. Convert to polar form if necessary.

2. Subtract x-components to get the x-component of the resultant vector.

3. Subtract y-components to get the y-component of the resultant vector.

4. If you want the resultant in polar form, convert from component to polar form.

Example:

Bingo, the physics dog, walks 50 meters from his doghouse in a straight line at a direction of 60(. Bingo then walks 75 meters in a straight line at a direction of 135(. How far and in what direction is Bingo from his doghouse?

Ans. 100 m, 106(

Example:

Bingo initially is 20 meters from his doghouse at direction of 0(. At a later time, Bingo is 20 meters from his house at a direction of -45(. Find Bingo’s displacement.

Ans. 15.3 m, 247.5(

Multiplying Vectors by Scalars

Component Form

1. Multiply each component by the scalar.

Polar Form

1. Multiply the magnitude by the absolute value of the scalar.

2. If the scalar is positive, the angle is unchanged. If the scalar is negative, add 180( to the angle.

1.3 1-D Translational Motion

Motion Quantities

The following quantities are components of 1-D vectors. These components can be positive or negative. The mks units of the quantities are given in brackets.

Position: x [m]

Displacement: (x = xf – xi [m]

Velocity*: [pic] [m/s]

Acceleration: [pic] [m/s/s = m/s2]

*The absolute value of the velocity is the speed (how fast). The sign of the velocity tells you the direction. If the sign is “+”, the object is moving to the right (+x direction). If the sign is “-“, the object moves to the left (-x direction).

Example:

Refer to the Average Velocity & Acceleration worksheet handed out in class.

Motion Plots

Motion plots are plots of position vs. time, velocity vs. time, and acceleration vs. time.

The slope of a line segment between two times on the position plot is the average velocity between those two times.

The slope of a line segment between two times on the velocity plot is the average acceleration between those two times.

Instantaneous Velocity and Acceleration

As the time interval (t shrinks towards zero, the motion plots become smooth as the line segments between the data points shrink.

The slope of the line that is tangent to the position curve at some time is the velocity at that time.

The slope of the line that is tangent to the velocity curve at some time is the acceleration at that time.

Given a motion plot, you can sketch the other two motion plots knowing the above relationships.

Example:

Refer to the Motion Graph worksheet handed out in class.

The above relationships can be expressed using the derivative and antiderivative as follows.

[pic]

[pic]

Example:

Suppose that the velocity of an object traveling in one dimension between t =0 and t = 4 seconds is given by the function vx(t) = 18 -3t2 where vx is in centimeters per second when t is in seconds.

a) Does the object stop ever stop during these four seconds? If so, when?

b) How fast and in which direction is the object moving at t = 2 seconds?

c) Find the object’s acceleration at t = 2 seconds.

d) Where is the object at t = 2 seconds if it is at x = 10 cm at t = 1 second?

Ans. (a) Yes, at t =2.45 s (b) 6 cm/s to the right (c) -12 cm/s2 (d) x = 21 cm

Equations of Motion for Constant Acceleration

[pic]

[pic]

[pic]

[pic]

Example:

An object, initially at rest, travels down a hill. The object speeds up uniformly, gaining 1.2 m/s each second.

a) How far does the object travel in the first three seconds?

b) How fast is it moving after these first three seconds?

c) How long does it take for the object to travel 20 meters to the bottom of the hill?

d) What is its speed at the bottom of the hill?

Ans. (a) 5.4 m (b) 3.6 m/s (c) 5.77 s (d) 6.9 m/s

Example:

A train is traveling at 40 m/s. When it reaches a flashing signal, it begins to slow down at a constant rate of 0.75 m/s/s.

a) How long does it take for the train to travel 1 km beyond the signal?

b) What is the train’s speed then?

Ans. (a) 40 s (b) 10 m/s

Free Fall Equations

We take up to be the positive direction and down to be the negative direction. The acceleration due to gravity near the Earth is ay = -g = -9.8 m/s2 = -32 ft./s2. Note that the symbol g represents a positive value.

[pic]

[pic]

[pic]

[pic]

Example:

An object is launched straight up at 20 m/s. Neglect air drag and answer the following questions.

a) How long does it take to reach the maximum height?

b) How high is this point above the launch height?

c) How long is the total flight time?

d) What is the impact speed of the object?

Ans. (a) 2.04 s (b) 20.4 m (c) 4.08 s (d) 20 m/s

Example:

A physics student is standing on a building holding a physics book 30 m above the ground. He tosses the book upwards. The book hits the sidewalk with a speed of 25 m/s. Air drag is negligible.

a) With what speed did he release the book?

b) How high does the book go?

c) How long is the book in the air?

Ans. (a) 6.08 m/s (b) 31.9 m above the sidewalk (c) 3.17 s

Example:

A monkey tosses a banana upwards at 5 m/s to another monkey that is sitting on a branch 6 meters above. How long does it take for the banana to reach the monkey in the tree? Air drag is negligible.

Ans. The banana doesn’t reach the monkey in the tree resulting in two frustrated monkeys.

Example:

The monkey in the previous example now tosses the banana upwards at 15 m/s. How long does it take now for the monkey in the tree to catch the banana?

Ans. If the monkey catches the banana on its way up, it takes 0.47 seconds. If caught on the way down, it takes 2.59 seconds.

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y

x

r

(

r1

r2rees (2-D) or angl direction. The magnitude has the units2

r1

r2rees (2-D) or angl direction. The magnitude has the units2

R = r1 + r22rees (2-D) or angl direction. The magnitude has the units

r1

r2rees (2-D) or angl direction. The magnitude has the units2

r1

R = r1 - r22rees (2-D) or angl direction. The magnitude has the units

-+rees (2-D) or angl direction. The magnitude has the unitsr2rees (2-D) or angl direction. The magnitude has the units2

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