Kinematics & Dynamics: Straight-Line Acceleration

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Kinematics & Dynamics: Straight-Line Acceleration In our discussion of vectors we talked about 2 different kinds of straight lines....what were they? 1) 2) What do we know about the relationship between the 2 kinds of vectors?

A. Calculate uniform and average acceleration. Acceleration is the rate at which velocity increases or decreases.

acceleration = ∆v ∆v = v2 – v1 ∆t ∆t = t2 – t1

SI units: m/s/s or m/s2.

A negative acceleration can mean 2 things. 1) The object moves in the ________________ direction with a decreasing speed (aka ____________________). 2) The object moves in the opposite (_________________) direction with an increasing speed (which would not

be referred to as deceleration).

o Sample problems: 1. An object has an initial velocity of 4 m/s and a final velocity of 37 m/s. Determine the acceleration

if the time interval was 6 s.

2. Determine the acceleration of an object with initial velocity of 70.0 m/s and final velocity of 30.0 m/s over a time interval of 14 s.

3. A ball is dropped on another planet. Determine the acceleration of a ball if it is initially falling at 3.0 m/s [down] and it ends with a final velocity of 10.0 m/s [down] over a time interval of 6.0s.

B. Formulas for Acceleration: Identify and solve for variables in acceleration problems. There are five variables for every situation where acceleration occurs: v

1, v

2, a, ∆t, ∆d

v = d/t is a CONSTANT SPEED formula. This CANNOT be used for an acceleration problem!!!

The formula for acceleration can be changed to produce:

v2 = v

1 + a∆t ∆d = v

1∆t + 1/2a(∆t)2 ∆d = v

2∆t - 1/2a(∆t)2 ∆d = 1/2(v

1 + v

2)∆t v

2

2 = v1

2 + (2ad)

Tips:

o Try to visualize the actual situation.

o List all the variables that are given.

o “?” the one(s) you are looking for.

o “X” the ones that are not given or at question.

o Find the formula that has the variables you have/need

o Use one or more extra digits in subsequent

calculations; round the answers to write your final

answer.

ACCELERATION ∆d = v2∆t - 1/2a∆t2 ∆d = v1∆t + 1/2a∆t2

a = ∆v/∆t = v2-v1/∆t ∆d = 1/2(v1 + v2) ∆t v2 = v1 + a∆t v2

2 = v12 + 2ad ag = g = 9.81 m/s2

NOTE: Acceleration can be a scalar or a vector quantity depending on if speed or velocity is provided.

CONSTANT SPEED v = ∆d/∆t

�⃑�av = ∆𝑑total/∆t

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1) A car moving at 15 m/s decelerates with a constant acceleration, coming to a rest in 6.0 s. What distance does the

car travel?

“At rest” or “stopped” means

An object that “decelerates” has a acceleration.

“How long” means and “how far” means

a =

v1 =

v2 =

∆d =

∆t =

2) If you drop a golf ball, how far does it fall in 0.50 s?

FOR ANY OBJECT IN THE VERTICAL (y) DIRECTION, the acceleration due to gravity is

An object that falls has an initial speed of

The final speed is ; we calculate the speed just before impact.

The displacement of a falling object is

a =

v1 =

v2 =

∆d =

∆t =

3) A pitcher throws a baseball straight up with a speed of 27 m/s. How long does the ball take to

reach its peak? What is its hang time?

The “peak” is the height where the object stops rising.

Speed is _____________ at the peak.

An object that is thrown upward has an initial speed (not 0 m/s).

“Hang time” is the time for an object to travel up and down.

The time to rise equals the time to fall (if the object starts and stops at the same point).

a =

v1 =

v2 =

∆d =

∆t =

4) A steel ball, starting from rest, rolls down one slope and up another. It takes 2.5 s to reach the bottom of the

first slope, at which point its speed is 5.0 m/s. If the magnitude of the acceleration on the second slope is

exactly one-half that on the first slope, how long will it take for the ball to come to a stop on the second slope?

HINT: Think about the PHYSICS of the problem. Draw a diagram and label your variables.

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Practice Problems: ***Note: Acceleration has both magnitude AND direction. What does your negative mean?

1) A cyclist accelerates from 5.0 m/s [S] to 15 m/s [S] in 4.0 s. What is his acceleration? (2.5 m/s2 [S])

2) A driver entering the outskirts of a city takes her foot off the accelerator so that her car slows down from 90.0 km/h to the speed limit. The average deceleration is 4.0 km/h/s over 10.0 s. Find the speed limit. (5.0 x 101 km/h)

3) A bullet is fired straight up with a muzzle velocity of 460 m/s. How long will it take it to reach its highest point? (47 s)

4) A dragster accelerates uniformly from rest to 56 m/s in 3.7 s. What is its acceleration? (15 m/s2)

5) A car decelerates at -2.0 m/s2. If its initial velocity is 24 m/s, to the east, what will its velocity be 8.0 s later? (8.0 m/s [E])

6) An arrow is shot straight up into the air at 40.0 m/s. As it falls back to Earth, the final velocity is 40.0 m/s [down]. How long does take the arrow to come back to the ground? (8.15 s) Hint: Think about the PHYSICS of this problem. You may want to draw a diagram and label your variables.

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7) Pressing on the brake pedal slows a car down from a velocity of 35 m/s to 20.0 m/s in 8.0 s. Assuming that the acceleration is uniform, what is the displacement of the car? (220 m [forward] OR 2.2 x 102 m [forw])

8) A balloon is descending at 4.0 m/s. An object is dropped from the descending balloon and lands on the ground 10.0 s later. What was the altitude (height) of the balloon at the moment the object was dropped? (530 m) NOTE: In your calculation you will get a negative number BUT the question says “what was the altitude (how high)?” so don’t write “[down]” for your answer.)

Extra Problem:

A ball thrown vertically upward returns to the ground 6.0 s later. a) How many seconds did the ball take to reach its highest point? (3.0 s)

b) How high did the ball go? (44 m)

c) With what velocity did it hit the ground? (29 m/s [down])

Challenge Yourself:

1) An object starts from rest and accelerates at 3.0 m/s2 for 4.0 s. its velocity remains constant for 7.0 s, and it finally comes to rest with uniform deceleration after another 5.0 s. Find a) the displacement for each stage of the motion (24 m, 84 m, 3.0 x 101 m) b) the average velocity over the whole time interval (8.6 m/s)

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Kinematics: Projectile Motion A. Identify variables for projectiles that are launched horizontally.

o Projectile -Any object which is projected by some means and continues to move due to its own ___________________ (mass).

Since a projectile moves in 2-dimensions, it therefore has 2 __________________, simultaneous one-dimensional motions at right angles to each other - Horizontal and Vertical components of the motion.

The horizontal component of the motion of a projectile is motion with constant _______________.

The vertical component of the motion of a projectile is motion with constant ____________________.

****Horizontal and Vertical Components must be considered independently!!*****

Horizontal “Velocity” Component

________________ changes, covers equal displacements in equal time periods. This means the initial horizontal velocity equals the final horizontal velocity WHY??

Vertical “Velocity” Component

Changes (due to ___________________), does NOT cover equal displacements in equal time periods.

Together, these components produce what is called a ________________ or path. This path is _________________ in nature.

Horizontally Launched Projectiles

The object is moving horizontally so vy is _______________ and any initial velocity that is given is the vx.

The time the object travels in the x-direction equals the time the object travels in the y-direction.

The vertical displacement of the object is _______________________.

To analyze a projectile in 2 dimensions we need 2 equations. One for the “x” direction and one for the “y” direction.

Remember, the time the object travels in the x-direction equals the time the object travels in the y-direction.

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o Sample Problems: 1. A stone is thrown horizontally at 15 m/s from the top of a cliff 44 m high. Neglecting air resistance,

a) Sketch the trajectory of the stone and label with relevant information.

b) How long does the stone take to reach the bottom of the cliff?

c) What is the range of the projectile? In other words, how far from the base of the cliff does the stone strike the ground?

d) What are the horizontal and vertical components of its velocity when it strikes?

e) What is its velocity of impact?

Note: What does a negative value indicate? When do you use a negative in your

calculations?

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2. A plane traveling with a horizontal velocity of 100.0 m/s is 500.0 m above the ground. At some point the pilot decides to drop some supplies to designated target below.

a) Sketch the trajectory of the supplies and label with relevant information.

b) How long is the drop in the air?

c) How far away from point where it was launched will it land?

d) What are the horizontal and vertical components of its velocity when it strikes?

e) What is its velocity of impact?

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Practice Problems:

1) A stone thrown horizontally from the top of a tall building takes 7.56 s to reach the street. How high is the building? (2.80 x 102 m) NOTE: In your calculation you will get a negative number BUT the question says “how high” so don’t write “[down]” for your answer.)

2) A bullet is fired horizontally at 300.0 m/s from a height of 1.5 m. Ignoring air resistance, calculate how far it travels horizontally before it hits the ground (Assume that the ground is level.) (170 m)

3) An object is projected horizontally with a velocity of 30.0 m/s. It takes 4.0 s to reach the ground. Neglecting air resistance, determine a) The height at which the object was projected (78 m)

b) The magnitude of the resultant velocity, just before the object strikes the ground (49 m/s)

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4) A bomber in level flight, flying at 92.0 m/s, releases a bomb at a height of 1950 m a) How long is it before the bomb strikes the Earth? (19.9 s)

b) How far does it travel horizontally? (1830 m)

c) What are the horizontal and vertical components of its velocity when it strikes? (92.0 m/s, 196 m/s)

d) What is its velocity of impact? (216 m/s [64.8°to the horizontal])

5) A cannonball shot horizontally from the top of a cliff with an initial velocity of 425 m/s is aimed towards a schooner on the ocean below. If the cliff is 78 m above the ocean surface, calculate the following:

a) The time for the cannonball to reach the water (4.0 s)

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b) The horizontal displacement of the cannonball (1700 m)

c) The velocity of the cannonball just before it strikes the water (430 m [5.4° to the horizontal])

B. Determine variables for projectile that are launched at an angle.

o Lab: Projectile Motion Simulation Lab o Projectiles on an Angle (Angle Launched)

o

The motion along a parabola is symmetrical; a vertical line through position C is the plane of symmetry.

All velocities have an x- and y-component except the velocity at the peak (C); the object stops rising, so the vy is _________________________.

The time taken for the object to travel from the initial to the peak position is ___________________ of the total time.

The initial vy equals the final vy but _____________________ are opposite.

The vertical displacement of the object is _____________________.

Vertically Launched Projectiles o Since the projectile was launched at an angle, the velocity MUST be broken into

components!!!

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There are several things you must consider when doing these types of projectiles besides using components. If it begins and ends at ground level, the “y” displacement is ZERO: y = 0

You will still use the same equations as for horizontally launched projectiles but you MUST USE COMPONENTS in the equation.

o Sample Problems: 1. A place kicker kicks a football with a velocity of 20.0 m/s and at an angle of 53 degrees.

Find the x and y components of the vector:

(a) How long is the ball in the air? (b) How far away does it land? (c) How high does it travel?

2. A gun shoots a bullet at 1200 m/s at an angle of 60.0 above the horizontal. Neglecting air resistance, determine

Find the x and y components of the vector:

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a) its time of flight

b) its range

c) its peak height

d) its velocity as it strikes the ground.

Practice Problems: 1) A projectile is launched with an initial speed of 21.8 m/s at an angle of 35.0-degrees above the horizontal.

Find the x and y components of the vector: (vx = 17.9 m/s; vy = 12.5 m/s) a) Determine the time of flight of the projectile. (2.55 s)

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b) Determine the peak height of the projectile. (7.96 m)

c) Determine the horizontal displacement of the projectile. (45.6 m)

d) Determine the resultant velocity of the projectile.

2) A projectile is launched with an initial speed of 30.0 m/s at an angle of 50.0-degrees above the horizontal. Find the x and y components of the vector: (vx = 19.3 m/s; vy = 23.0 m/s) a) Determine the time of flight of the projectile. (4.69 s)

b) Determine the peak height of the projectile. (27.0 m)

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c) Determine the horizontal displacement of the projectile. (90.5 m)

d) Determine the resultant velocity of the projectile.

Extra Problems: 3) A cannonball is fired with a velocity of 100.0 m/s 25° above the horizontal. Determine how far away it lands on

level ground. (7.8 x 102 m) 4) A baseball player throws a ball at 35.0 m/s, what is its maximum range? (125 m)

(Hint: From your lab, which angle produces the maximum range)

Challenge Yourself: 5) A player kicks a soccer ball towards the goalkeeper, at an angle of 37° to the horizontal and with an initial speed

of 14.7 m/s. The goalkeeper stands 26.0 m from the kicker. Where will the ball land relative to the goalkeeper? (4.8 m short)

6) In problem 5, the goalkeeper watches the ball until it reaches maximum height, then runs at constant speed to just

intercept it at ground level. How fast must he run? (5.3 m/s)