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# Chapter 5: Basic Physics SE-Motion

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The Big Idea

Speed represents how quickly an object is moving through space. Velocity is speed with a direction, making it a vector quantity. If an object’s velocity changes with time, the object is said to be accelerating. As we’ll see in the next chapters, understanding the acceleration of an object is the key to understanding its motion. We will assume constant acceleration throughout this chapter.

Key Definitions

Vectors

$x =$ position (m)

$\Delta x =$ displacement $= x_f - x_i$

$v =$ velocity (m/s)

$v_i =$ initial velocity

$v_f =$ final velocity

$\Delta v =$ change in velocity $= v_f - v_i$

$a =$ acceleration $(m/s^2)$

Scalars

$t =$ time (s)

$d =$ distance $(m) = | \Delta x_1 | + | \Delta x_2 | + \ldots$

$v =$ speed (m/s) $= | v |$

Symbols

$\Delta \text{(anything)} = \text{final value} - \text{initial value}$

Key Equations

• $v_{avg} = \frac{\Delta x}{\Delta t}$
• $a_{avg} = \frac{\Delta v}{\Delta t}$

The Big Three

• $\Delta x = v_it + \frac{1}{2} at^2$ ; allows you to calculate the displacement at some time $t$.
• $\Delta v = at$ ; allows you to calculate the velocity after some time $t$.
• $v_f{^2} = v_i{^2} + 2a( \Delta x)$ ; a combination of previous two equations.

Key Concepts

• When you begin a problem, define a coordinate system. For positions, this is like a number line; for example, positive $(+x)$ positions can be to the right of the origin and negative $(-x)$ positions to the left of the origin.
• For velocity $v$ you might define positive as moving to the right and negative as moving to the left. What would it mean to have a positive position and a negative velocity?
• For acceleration $a$, you might define positive as caused by a force to the right and negative as a force to the left. What would it mean to have a negative velocity and a positive acceleration? Careful, positive acceleration does not always mean increasing speed!
• Be sure you understand the difference between average velocity (measured over a long period of time) and instantaneous velocity (measured at a single moment in time).
• Gravity near the Earth pulls an object downwards toward the surface of the Earth with an acceleration of $9.8 \ m/s^2 ( \approx 10 \ m/s^2)$. In the absence of air resistance, all objects will fall with the same acceleration.
• Deceleration is the term used when an object’s speed is decreasing due to acceleration in the opposite direction of its velocity.
• If there is constant acceleration the graph $x$ vs. $t$ produces a parabola. The slope of the graph equals the instantaneous velocity. The slope of a $v$ vs. $t$ graph equals the acceleration.
• The slope of the graph $v$ vs. $t$ can be used to find acceleration; the area of the graph $v$ vs. $t$ can be used to find displacement. Welcome to calculus!
• At first, you might get frustrated trying to figure out which of the Big Three equations to use for a certain problem, but don’t worry, this comes with practice. Making a table that identifies the variables given in the problem and the variables you are looking for can sometimes help.

Solved Examples

Example 1: Pacific loggerhead sea turtles migrate over 7,500 miles (12,000 km) between nesting beaches in Japan and feeding grounds off the coast of Mexico. If the average speed of a loggerhead is about 45 km/day, how long does it take for it to complete a one-way migration?

Question: $t = ?$ [days]

Given: $d = 12,000 \ km$

${\;} \qquad v_{avg} = 45 \ km/day$

Equation: $v_{avg} = \frac{d}{t}$ therefore $t = \frac{d}{v_{avg}}$

Plug n’ Chug: $t = \frac{d}{v_{avg}} = \frac{12,000 \ km}{45 \ km/day} = 267 \ days$

Answer: $\boxed{\mathbf{267 \ days}}$

Example 2: A Top Fuel dragster can accelerate from 0 to 100 mph (160 km/hr) in 0.8 seconds. What is the average acceleration in $m/s^2$?

Question: $a_{avg} = ? \ [m/s^2]$

Given: $v_i = 0 \ m/s$

${\;} \qquad \ \ v_f = 160 \ km/hr$

${\;} \qquad \ \quad t = 0.8 \ s$

Equation: $a_{avg} = \frac{\Delta v }{t}$

Plug n’ Chug: Step 1: Convert km/hr to m/s

$v_f = \left( 160 \frac{km}{hr} \right ) \left( \frac{1,000 \ m}{1 \ km} \right ) \left ( \frac{1 \ hr}{3,600 \ s} \right ) = 44.4 \ m/s$

Step 2: Solve for average acceleration:

$a_{avg} = \frac{\Delta v}{t} = \frac{v_f - v_i}{t} = \frac{44.4 \ m/s - 0 \ m/s}{0.8 \ s} = 56 \ m/s^2$

Answer: $\boxed {\mathbf{56 \ m/s^2}}$ Note that this is over $5 \frac{1}{2}$ g’s!

Example 3: While driving through Napa you observe a hot air balloon in the sky with tourists on board. One of the passengers accidentally drops a wine bottle and you note that it takes 2.3 seconds for it to reach the ground. (a) How high is the balloon? (b) What was the wine bottle’s velocity just before it hit the ground?

Question a: $h = ? [m]$

Given: $t = 2.3 \ s$

${\;} \qquad \quad g = 10 \ m/s^2$

${\;} \qquad \quad v_i = 0 \ m/s$

Equation: $\Delta x = v_it + \frac{1}{2}at^2$ or $h = v_it + \frac{1}{2}gt^2$

Plug n’ Chug: $h = 0 + \frac{1}{2}(10 \ m/s^2)(2.3 \ s)^2 = 26.5 \ m$

Answer: $\boxed{\mathbf{26.5 \ m}}$

Question b: $v_f = ? [m/s]$

Given: (same as above)

Equation: $v_f = v_i + at$

Plug n’ Chug: $v_f = v_i +at = 0 + (10 \ m/s^2)(2.3 \ s) = 23 \ m/s$

Answer: $\boxed{\mathbf{23 \ m/s}}$

Example 4: The second tallest building in the world is the Petronas Tower in Malaysia. If you were to drop a penny from the roof which is 378.6 m (1242 ft) high, how long would it take to reach the ground? You may neglect air friction.

Question: $t = ? [s]$

Given: $h = 378.6 \ m$

${\;} \qquad \quad g = 10 \ m/s^2$

${\;} \qquad \quad v_i = 0 \ m/s$

Equation: $\Delta x = v_it + \frac{1}{2}at^2$ or $h = v_it + \frac{1}{2}gt^2$

Plug n’ Chug: since $v_i = 0$, the equation simplifies to $h = \frac{1}{2} gt^2$ rearranging for the unknown variable, $t$, yields

$t = \sqrt{\frac{2h}{g}} = \sqrt{\frac{2(378.6 \ m)}{10.0 \ m/s^2}} = 8.70 \ s$

Answer: $\boxed{\mathbf{8.70 \ s}}$

One-Dimensional Motion Problem Set

1. A car drives around the circle in front of Stent Hall at 5 mph. Is it moving with a constant speed? Constant velocity? Is it accelerating? Explain.
2. Answer the following questions about one-dimensional motion. (a) What is the difference between distance $d$ and displacement $\Delta x ?$ Write a sentence or two explaining this and give an example of each. (b) Does the odometer reading in a car measure distance or displacement? (c) Imagine a fox darting around in the woods for several hours. Can the displacement $\Delta x$ of the fox from his initial position ever be larger than the total distance $d$ he traveled? Explain. (d) What is the difference between acceleration and velocity? Write a paragraph that would make sense to a $5^{th}$ grader. (e) Give an example of a situation where an object has an upward velocity but a downward acceleration. (f) What is the difference between average and instantaneous velocity? Make up an example involving a trip in a car that demonstrates your point. (g) If the position of an object is increasing linearly with time (i.e., $\Delta x$ is proportional to $t$), what can we say about its acceleration? Explain your thinking. (h) If the position of an object is increasing non-linearly with time (i.e., $\Delta x$ is not proportional to $t$), what can we say about its velocity? Explain your thinking for each graph below.
3. A cop passes you on the highway. Which of the following statements must be true at the instant he is passing you? You may choose more than one answer.
1. Your speed and his speed are the same.
2. Your position $x$ along the highway is the same as his position $x$ along the highway.
3. Your acceleration and his acceleration are the same.
4. If a car is slowing down from 50 MPH to 40 MPH, but the $x$ position is increasing, which of the following statements is true? You may choose more than one.
1. The velocity of the car is in the $+x$ direction.
2. The acceleration of the car is in the same direction as the velocity.
3. The acceleration of the car is in the opposite direction of the velocity.
4. The acceleration of the car is in the $-x$ direction.
5. A horse is galloping forward with an acceleration of $3 \ m/s^2$. Which of the following statements is necessarily true? You may choose more than one.
1. The horse is increasing its speed by 3 m/s every second, from 0 m/s to 3 m/s to 6 m/s to 9 m/s.
2. The speed of the horse will triple every second, from 0 m/s to 3 m/s to 9 m/s to 27 m/s.
3. Starting from rest, the horse will cover 3 m of ground in the first second.
4. Starting from rest, the horse will cover 1.5 m of ground in the first second.
6. Below are images from a race between Ashaan (above) and Zyan (below), two daring racecar drivers. High speed cameras took four pictures in rapid succession. The first picture shows the positions of the cars at $t = 0.0$. Each car image to the right represents times 0.1, 0.2, and 0.3 seconds later. (a) Who is ahead at $t = 0.2 \ s$? Explain. (b) Who is accelerating? Explain. (c) Who is going faster at $t = 0.3 \ s$? Explain. (d) Which car has a constant velocity throughout? Explain. (e) Graph $x$ vs. $t$ and $v$ vs. $t$. Put both cars on same graph; label which line is which car. (f) Who is going faster at $t = 0.2 \ s?$ Justify your answer (this one is a thinker!)
7. The position graph below is of the movement of a fast turtle who can turn on a dime.
1. Sketch the velocity vs. time graph of the turtle below.
2. Explain what the turtle is doing (including both speed and direction) from
1. 0-2s
2. 2-3s
3. 3-4s
3. How much distance has the turtle covered after 4s?
4. What is the turtle’s displacement after 4s?
8. Draw the position vs. time graph that corresponds to the velocity vs. time graph below. You may assume a starting position $x_0 = 0$. Label the $y-$axis with appropriate values.
9. The following velocity-time graph represents 10 seconds of actress Halle Berry’s drive to work (it’s a rough morning).

(a) Fill in the tables below – remember that displacement and position are not the same thing!

Instantaneous Time (s) Position (m)
Interval (s) Displacement (m) Acceleration $(m/s^2)$ 0 sec 0 m
0-2 sec
2 sec
2-4 sec
4 sec
4-5 sec
5 sec
5-9 sec
9 sec
9-10 sec
10 sec

(b) On the axes below, draw an acceleration-time graph for the car trip. Include numbers on your acceleration axis.

(c) On the axes below, draw a position-time graph for the car trip. Include numbers on your position axis. Be sure to note that some sections of this graph are linear and some curve – why?

1. Two cars are drag racing down El Camino. At time $t = 0$, the yellow Maserati starts from rest and accelerates at $10 \ m/s^2$. As it starts to move it’s passed by a ’63 Chevy Nova (cherry red) traveling at a constant velocity of 30 m/s. (a) On the axes below, show a line for each car representing its speed as a function of time. Label each line. (b) At what time will the two cars have the same speed (use your graph)? (c) On the axes below, draw a line (or curve) for each car representing its position as a function of time. Label each curve. d. At what time would the two cars meet (other than at the start)?
2. Two cars are heading right towards each other but are 12 km apart. One car is going 70 km/hr and the other is going 50 km/hr. How much time do they have before they collide head on?
3. A cheetah can accelerate at almost one “$g$”, or $10 \ m/s^2$, for short periods of time. Suppose a cheetah sees a gazelle and accelerates from rest for 2.8 s at $10 \ m/s^2$.
1. Calculate the cheetah’s final velocity.
2. Calculate the cheetah’s average velocity.
3. Calculate the cheetah’s displacement.
4. Sketchy LeBaron, a used car salesman, claims his car is able to go from 0 to 60 mi/hr in 3.5 seconds.
1. Convert 60 mi/hr to m/s. Show your work!
2. What is the average acceleration of this car? Give your answer in $m/s^2$. (Hint: you will have to perform a conversion.)
3. What’s the car’s average velocity for this time period? How much distance does this car cover in these 3.5 seconds? Express your answer twice: in meters and in feet.
4. What is the speed of the car in mi/hr after 2 seconds?
5. A car is smashed into a wall during Weaverville’s July $4^{th}$ Destruction Derby. The car is going 25 m/s just before it strikes the wall. It comes to a stop 0.8 seconds later. What is the average acceleration of the car during the collision?
6. You are speeding towards a brick wall at a speed of 55 MPH. The brick wall is only 100 feet away.
1. What is your speed in m/s?
2. What is the distance to the wall in meters?
3. What is the minimum acceleration you should use to avoid hitting the wall?
7. You throw a ball straight up into the air. At the top of its trajectory, what is its instantaneous velocity? Its instantaneous acceleration? Explain why these values are not the same.
8. You drop a rock from the top of a cliff. The rock takes 3.5 seconds to reach the bottom.
1. What is the initial speed of the rock?
2. What is the acceleration of the rock at the moment it is dropped?
3. How fast is the rock going half-way (in time) down?
4. What is the acceleration of the rock when it is half-way down the cliff?
5. How fast is the rock traveling when it hits bottom?
6. What is the height of the cliff?
1. Convert this height into meters.
2. Assuming no air resistance, at what speed did he leave the ground?
3. What is his speed $\frac{3}{4}$ of the way up?
4. What is his speed just before he hits the ground on the way down?
10. What acceleration should you use to increase your speed from 10 m/s to 18 m/s over a distance of 55 m?
11. You are standing on a balcony on the $15^{th}$ floor of the Bank of America building in San Francisco. The balcony is 72 m above the ground; the top of the building is 52 floors (237 m) high. You launch a ball straight up in the air from the balcony. The initial vertical speed is 75 m/s. (For this problem, you may ignore your own height, which is very small compared to the height of the building.)
1. How high up does the ball go above the ground?
2. After the ball has been in the air for 4.2 sec, calculate
1. its instantaneous velocity at 4.2 s . Is it rising or falling? How can you tell?
2. its average velocity for the first 4.2 s.
3. its position at 4.2 s. Is it above or below the balcony? How can you tell?
3. How fast is the ball going right before it reaches the top of the building?
4. For how many seconds total is the ball in the air (assuming he catches it on balcony–ouch!)?
12. Measure how high you can jump vertically on Earth. Then, figure out how high you would be able to jump on the Moon, where acceleration due to gravity is $1/6^{th}$ that of Earth. Assume you launch upwards with the same speed on the Moon as you do on the Earth.
13. You drive to San Francisco along 101 at 60 mph (30 miles north your home). On the return trip, there is traffic on 101 (next time take Caltrain...), so you are only going 30 mph.
1. How long does it take you while traveling to San Francisco?
2. How long does it take you while traveling back home?
3. Calculate your average speed for the whole trip (Hint: the answer is NOT 45 mph).

Answers (answers assume $10 \ m/s^2$ for acceleration of gravity):

1. discuss in class
2. discuss $n$ class
3. b
4. a,c,d
5. a,d
6. (a) Zyan (b) Ashaan is accelerating (c) Ashaan (d) Zyan (f) Ashaan
7. (c) 25 m (d) -5 m
8. discuss in class
9. discuss in class
10. (b) 3 sec (d) 6 sec
11. 6 min.
1. 28 m/s
2. 14 m/s
3. 39.2 m
1. 26.8 m/s
2. $7.7 \ m/s^2$
3. 13.4 m/s, 46.9 m
4. 34 mph
12. $-31 \ m/s^2$
1. 25 m/s
2. 30.5 m
3. $10.2 \ m/s^2$
13. discuss in class
1. 0 m/s
2. $-10 \ m/s^2$
3. -17.5 m/s
4. $-10 \ m/s^2$
5. -35 m/s
6. 61 m
1. 1.22 m
2. 4. m/s
3. 2.47 m/s
4. -4.94 m/s
14. $2 \ m/s^2$
1. 353 m
2. 33 m/s rising; 54 m/s, 227 m above balcony
3. 48.2 m/s
4. 15.0 s
15. 6 times higher
1. 0.5 hr
2. 1 hr
3. 40 mph

OPTIONAL PROBLEMS

1. In the picture to the right, a ball starting at rest rolls down a ramp, goes along at the bottom, and then back up a smaller ramp. Ignore friction and air resistance. Sketch the vertical position vs. time and vertical speed vs. time graphs that accurately describe this motion. Label vertical axes of your graphs.
2. You are sitting on your bike at rest. Your brother comes running at you from behind at a speed of 2 m/s. At the exact moment he passes you, you start up on your bike with an acceleration of $2 \ m/s^2$. (a) Draw a picture of the situation, defining the starting positions, speeds, etc. (b) At what time $t$ do you have the same speed as your brother? (c) At what time $t$ do you pass your brother? (d) Draw another picture of the exact moment you catch your brother. Label the drawing with the positions and speeds at that moment. (e) Sketch a position vs. time graph for both you and your brother, labeling the important points (i.e., starting point, when you catch him, etc.) (f) Sketch a speed vs. time graph for both you and your brother, labeling the important points (i.e., starting point, when you catch him, etc.)
3. A helicopter is traveling with a velocity of 12 m/s directly upward. Directly below the helicopter is a very large and very soft pillow. As it turns out, this is a good thing, because the helicopter is lifting a large man. When the man is 20 m above the pillow, he lets go of the rope.
1. What is the speed of the man just before he lands on the pillow?
2. How long is he in the air after he lets go?
3. What is the greatest height reached by the man above the ground? (Hint: this should be greater than 20 m. Why?)
4. What is the distance between the helicopter and the man three seconds after he lets go of the rope?

1. discuss in class
2. (b) 1 sec (c) 2 sec (d) 4 m
1. 23 m/s
2. 3.5 s
3. 27.2 m
4. 45 m

Feb 23, 2012