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Skill Sheet 4.1
Acceleration Problems
This skill sheet will allow you to practice solving acceleration problems. Remember that acceleration is the rate of change in the speed of an object. In other words, at what rate does an object speed up or slow down? A positive value for acceleration refers to the rate of speeding up, and negative value for acceleration refers to the rate of slowing down. The rate of slowing down is also called deceleration. To determine the rate of acceleration, you use the formula: Final speed – Beginning speed Acceleration = ------------------------------------------------------------------------Change in Time
1. Solving acceleration problems Solve the following problems using the equation for acceleration. Remember the units for acceleration are meters per second per second or m/s2. The first problem is done for you. 1.
A biker goes from a speed of 0.0 m/s to a final speed of 25.0 m/s in 10 seconds. What is the acceleration of the bicycle? 25.0 m 25.0 m 0.0 m ------------------------------- – ------------s 2.5 ms s acceleration = ------------------------------------ = ---------------= -----------2 10 s 10 s s
2.
A skater increases her velocity from 2.0 m/s to 10.0 m/s in 3.0 seconds. What is the acceleration of the skater?
3.
While traveling along a highway a driver slows from 24 m/s to 15 m/s in 12 seconds. What is the automobile’s acceleration? (Remember that a negative value indicates a slowing down or deceleration.)
4.
A parachute on a racing dragster opens and changes the speed of the car from 85 m/s to 45 m/s in a period of 4.5 seconds. What is the acceleration of the dragster?
5.
The cheetah, which is the fastest land mammal, can accelerate from 0.0 mi/hr to 70.0 mi/hr in 3.0 seconds. What is the acceleration of the cheetah? Give your answer in units of mph/s.
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Investigation
6.
The Lamborghini Diablo sports car can accelerate from 0.0 km/hr to 99.2 km/hr in 4.0 seconds. What is the acceleration of this car? Give your answer in units of kilometers per hour/s.
7.
Which has greater acceleration, the cheetah or the Lamborghini Diablo? (To figure this out, you must remember that there are 1.6 kilometers in 1 mile.) Be sure to show your calculations.
2. Solving for other variables Now that you have practiced a few acceleration problems, you can rearrange the acceleration formula so that you can solve for other variables such as time and final speed. Final speed = Beginning speed + acceleration u time speed – Beginning speedTime = Final -----------------------------------------------------------------------Acceleration 1.
A cart rolling down an incline for 5.0 seconds has an acceleration of 4.0 m/s2. If the cart has a beginning speed of 2.0 m/s, what is its final speed?
2.
A car accelerates at a rate of 3.0 m/s2. If its original speed is 8.0 m/s, how many seconds will it take the car to reach a final speed of 25.0 m/s?
3.
A car traveling at a speed of 30.0 m/s encounters an emergency and comes to a complete stop. How much time will it take for the car to stop if its rate of deceleration is -4.0 m/s2?
4.
If a car can go from 0.0 to 60.0 mi/hr in 8.0 seconds, what would be its final speed after 5.0 seconds if its starting speed were 50.0 mi/hr?
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Skill Sheet 4.3
Acceleration Due to Gravity
One formal description of gravity is “The acceleration due to the force of gravity.” The relationships among gravity, speed, and time are identical to those among acceleration, speed, and time. This skill sheet will allow you to practice solving acceleration problems that involve objects that are in free fall.
1. Gravity, velocity, distance, and time When solving for velocity, distance, or time with an object accelerated by the force of gravity, we start with an advantage. The acceleration is known to be 9.8 meters/second/second or 9.8 m/sec2. However, three conditions must be met before we can use this acceleration: •
The object must be in free fall.
•
The object must have negligible air resistance.
•
The object must be close to the surface of the Earth.
In all of the examples and problems, we will assume that these conditions have been met and therefore acceleration due to the force of gravity shall be equal to 9.8 m/sec2 and shall be indicated by g Because the y-axis of a graph is vertical, change in height shall be indicated by y. Remember that speed refers to “how fast” in any direction, but velocity refers to “how fast” in a specific direction. The sign of numbers in these calculations is important. Velocities upward shall be positive, and velocities downward shall be negative.
2. Solving for velocity Here is the equation for solving for velocity: final velocity = initial velocity + the acceleration due to the force of gravity u time OR v = v 0 + gt Example: How fast will a pebble be traveling 3 seconds after being dropped? v = v 0 + gt 2
v = 0 + – 9.8 meters/s u 3 s v = – 29.4 meters/s (Note that gt is negative because the direction is downward.)
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Investigation
3. Problems 1.
A penny dropped into a wishing well reaches the bottom in 1.50 seconds. What was the velocity at impact?
2.
A pitcher threw a baseball straight up at 35.8 meters per second. What was the ball’s velocity after 2.5 seconds? (Note that, although the baseball is still climbing, gravity is accelerating it downward.)
3.
In a bizarre but harmless accident, Superman fell from the top of the Eiffel Tower. How fast was Superman traveling when he hit the ground 7.8 seconds after falling?
4.
A water balloon was dropped from a high window and struck its target 1.1 seconds later. If the balloon left the person’s hand at –5 meters/s, what was its velocity on impact?
4. Solving for distance Imagine that an object falls for one second. We know that at the end of the second it will be traveling at 9.8 meters/second. However, it began its fall at zero meters/second. Therefore, its average velocity is half of 9.8 meters/second. We can find distance by multiplying this average velocity by time. Here is the equation for solving for distance. Look to find these concepts in the equation: the acceleration due to the force of gravity u time distance = ---------------------------------------------------------------------------------------------------------------------- u time 2 OR 1 2 y = --- gt 2 Example: A pebble dropped from a bridge strikes the water in exactly 4 seconds. How high is the bridge? 1 2 y = --- gt 2 1 y = --- u 9.8 meters/sec u 4 s u 4 s 2 1 2 y = --- u 9.8 meters/s u 4 s u 4 s 2 y = 78.4 meters Note that the terms cancel. The answer written with the correct number of significant figures is 78 meters. The bridge is 78 meters high.
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Investigation
5. Problems 1.
A stone tumbles into a mineshaft and strikes bottom after falling for 4.2 seconds. How deep is the mineshaft?
2.
A boy threw a small bundle toward his girlfriend on a balcony 10.0 meters above him. The bundle stopped rising in 1.5 seconds. How high did the bundle travel? Was that high enough for her to catch it?
3.
A volleyball serve was in the air for 2.2 seconds before it landed untouched in the far corner of the opponent’s court. What was the maximum height of the serve?
6. Solving for time The equations demonstrated in Sections 2 and 3 can be used to find time of flight from speed or distance, respectively. Remember that an object thrown into the air represents two mirror-image flights, one up and the other down. Original equation Time from velocity Time from distance
Rearranged equation to solve for time v–v t = -------------0 g
v = v 0 + gt 1 2 y = --- gt 2
t =
2y -----g
Try these: 1.
At about 55 meters/s, a falling parachuter (before the parachute opens) no longer accelerates. Air friction opposes acceleration. Although the effect of air friction begins gradually, imagine that the parachuter is free falling until terminal speed (the constant falling speed) is reached. How long would that take?
2.
The climber dropped her compass at the end of her 240-meter climb. How long did it take to strike bottom?
3.
For practice and to check your understanding, use these equations to check your work in Sections 2 and 3.
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Skill Sheet 10.3
Potential and Kinetic Energy
In this skill sheet, you will review the forms of energy and formulas for two kinds of energy: potential and kinetic. After having worked through this skill sheet, calculating the amount of kinetic or potential energy for an object will be easy!
1. Forms of energy Energy can be used or stored. When talking about motion, energy that is stored is called potential energy. Energy that is used when an object is moving is called kinetic energy. Other forms of energy include radiant energy from the sun, chemical energy from the food you eat, and electrical energy from the outlets in your home. Energy is measured in joules or newton-meters. 2
m1 joule = 1 kg -----= 1 N m = 1 joule 2 s m 1 N = 1 kg ----2 s
2. Potential energy The word potential means that something is capable of becoming active. Potential energy sometimes is referred to as stored energy. This type of energy often comes from the position of an object relative to Earth. A diver on the high board has more potential energy than someone who dives into the pool from the low board. The formula to calculate the potential energy of an object is the mass of the object times the acceleration of gravity times the height of the object. E p = mgh The mass of the object times the acceleration of gravity (g) is the same as the weight of the object in newtons. The acceleration of gravity is equal to 9.8 m/sec2. 9.8 m - = weight of the object (newtons) mass of the object (kilograms) u -----------2 s
3. Kinetic energy The second category of energy is kinetic energy, the energy of motion. Kinetic energy depends on the mass of the object as well as the speed of that object. Just imagine a large object moving at a very high speed. You would say that the object has a lot of energy. Since the object is moving, it has kinetic energy. The formula for kinetic energy is: 1 2 E k = --- mv 2 To do this calculation, you need to square the velocity value. Next, multiply by the mass, and then divide by 2.
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Investigation
4. Solving problems Now you can practice calculating potential and kinetic energy. Make sure to show your work with all units present in your calculations as well as in your answer. Write your answers in joules. The first two problems have been done for you. 1.
A 50-kilogram boy and his 100-kilogram father went jogging. Both ran at a rate of 5 m/s. Who had more kinetic energy? Show your work and explain. Although the boy and his father were running at the same speed, the father has more kinetic energy because he has more mass. The kinetic energy of the boy: 2
m 5m 2 1 --- 50 kg § ---------· = 625 kg ------ = 625 joules 2 © ¹ s 2 s The kinetic energy of the father: 2
5m 2 1 m --- 100 kg § ---------· = 1 250 kg -----= 1,250 joules 2 © ¹ 2 s s 2.
What is the potential energy of a 10-newton book sitting on a shelf 2.5 meters high? The book’s weight (10 newtons) is equal to its mass times the acceleration of gravity. Therefore, you can easily use this value in the potential energy formula: potential energy = mgh = 10 N 2.5 m = 25 N m = 25 joules
3.
Determine the amount of potential energy of a 5-newton book that is moved to three different shelves on a bookcase. The height of the shelves is 1.0 meter, 1.5 meters, and 2.0 meters.
4.
Two objects were lifted by a machine. One object had a mass of 2 kilograms, and was lifted at a speed of 2 m/s. The other had a mass of 4 kilograms and was lifted at 3 m/s. Which object had more kinetic energy while it was being lifted? Show all calculations.
5.
In problem 4, which object had more potential energy when it was lifted a distance of 10 meters? Show your calculation. (Remember that gravity = 9.8 m/sec2)
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Investigation
6.
You are standing in your in-line skates at the top of a large hill. Your potential energy is equal to 1,000 joules. The last time you checked, your mass was 60 kilograms. a. What is your weight in newtons?
b. What is the height of the hill?
c. If you start skating downhill, your potential energy will be converted to kinetic energy. At the bottom of the hill, your kinetic energy will be equal to your potential energy at the top. What will be your speed at the bottom of the hill?
7.
Answer the following: a. A 1-kilogram ball is thrown into the air with an initial velocity of 30 m/s. How much kinetic energy does the ball have?
b. How much potential energy does the ball have when it reaches the top of its ascent?
c. How high into the air did the ball travel?
8.
What is the potential energy of a 3-kilogram ball lying on the ground?
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Investigation
9.
What is the kinetic energy of a 2,000-kilogram boat moving at 5 m/s?
10. What is the velocity of an 500-kilogram elevator that has 4,000 joules of energy?
11. What is the mass of an object that creates 33,750 joules of energy by moving at 30 m/s?
12. Challenge problem: In the diagram at right, the potential energy of the ball at position A equals its kinetic energy at position C. At position A, the ball has zero velocity so its kinetic energy equals zero. At position C, the ball does not have potential energy because its height is zero. The mass of the ball is 1 kilogram. Use this information to find the velocity of the ball at position B. a. Write an equation that shows how the energy of the ball at position B relates to its potential energy at position A.
b. What is the velocity of the ball at position B?
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Skill Sheet 19.3A
Ohm’s Law
Building and testing series circuits has helped you understand the relationship between voltage, resistance, and current. You know that if the voltage (energy) in a circuit increases, so does the current (flow of charges). You also understand that if the resistance increases, the current flow decreases. The German physicist Georg S. Ohm developed this mathematical relationship, which is present in most circuits. This relationship is known as Ohm’s law: Voltage Current = -----------------------Resistance This skill sheet will provide you with an opportunity to test your knowledge of Ohm’s law.
1. Using Ohm’s law to understand circuits As you use Ohm’s law, remember that the unit for current is amperes or amps. The unit for voltage is volts, and the unit for resistance is ohms (symbolized : . To work through this skill sheet, you will need the symbols used to depict circuits in diagrams. The symbols that are most commonly used for circuit diagrams are provided at right. All of the circuits discussed in this skill sheet are series circuits. This means the current has only one path through the circuit. Later, you will learn about another kind of circuit in which the current has more than one possible path. This type of circuit is called a parallel circuit. Note: For convenience, the symbol for battery is used to represent one or more batteries. The batteries you have used to build circuits are 1.5 -volt batteries. Dividing the total voltage by 1.5 volts will tell you the number of batteries present in the circuit. For example, the total voltage in the second diagram on the right is 6 volts. Divide 6 volts by 1.5 to find the number of batteries in the circuit (6 y 1.5 = 4). There are four batteries in the circuit.
2. Solving problems In this section, you will find some problems based on diagrams and others without diagrams. In all cases, you should show your work. 1.
If a toaster produces 12 ohms of resistance in a 120-volt circuit, what is the amount of current in the circuit?
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Investigation
2.
You have a large flashlight that requires 4 D-cell batteries. If the current in the flashlight is 2 amps, what is the resistance of the light bulb? (HINT: One D-cell battery has 1.5 volts.)
3.
What is the voltage of a circuit with 15 amps of current and a toaster with 8 ohms of resistance?
4.
Use the diagram below to answer the following problems:
a. What is the total voltage in each circuit?
b. If the bulbs are identical, which circuit has a greater current? Explain your answer.
c. How does the brightness of the bulb in circuit A compare to the brightness in circuit B? Explain your answer.
d. How much current would be measured in each circuit if the light bulb had a resistance of 6 ohms?
e. How much current would be measured in each circuit if the light bulb had a resistance of 12 ohms?
f. Suppose a second bulb is added to each of the circuits in series. Explain what would happen to the current in each circuit.
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Investigation
5.
Use the diagram below to answer the following problems:
a. How much current would be measured in each circuit if each light bulb had a resistance of 6 ohms?
b. How much current would be measured in each circuit if each light bulb had a resistance of 12 ohms?
c. What happens to the brightness of each bulb as you add bulbs to a series circuit? (HINT: Compare these diagrams to the ones in question 4 above.)
6.
What happens to the amount of current in a series circuit as the number of batteries increases?
7.
What happens to the amount of current in a series circuit as the number of bulbs increases?
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20.1B Series Circuits
In a series circuit, current follows only one path from the positive end of the battery toward the negative end. The total resistance of a series circuit is equal to the sum of the individual resistances. The amount of energy used by a series circuit must equal the energy supplied by the battery. In this way, electrical circuits follow the law of conservation of energy. Understanding these facts will help you solve problems that deal with series circuits. To answer the questions in the practice section, you will have to use Ohm's law. Remember that: Voltage (volts) Current (amps) = --------------------------------------Resistance (ohms) Some questions ask you to calculate a voltage drop. We often say that each resistor (or light bulb) creates a separate voltage drop. As current flows along a series circuit, each resistor uses up some energy. As a result, the voltage gets lower after each resistor. If you know the current in the circuit and the resistance of a particular resistor, you can calculate the voltage drop using Ohm’s law. Voltage drop across resistor (volts) = Current through resistor (amps) u Resistance of one resistor (ohms)
1.
2.
Use the series circuit pictured to the right to answer questions (a)-(e). a.
What is the total voltage across the bulbs?
b.
What is the total resistance of the circuit?
c.
What is the current in the circuit?
d.
What is the voltage drop across each light bulb? (Remember that voltage drop is calculated by multiplying current in the circuit by the resistance of a particular resistor: V = IR.)
e.
Draw the path of the current on the diagram.
Use the series circuit pictured to the right to answer questions (a)-(e). a.
What is the total voltage across the bulbs?
b.
What is the total resistance of the circuit?
c.
What is the current in the circuit?
d.
What is the voltage drop across each light bulb?
e.
Draw the path of the current on the diagram.
3.
What happens to the current in a series circuit as more light bulbs are added? Why?
4.
What happens to the brightness of each bulb in a series circuit as additional bulbs are added? Why?
20.1B Series Circuits
5.
6.
7.
8.
9.
Use the series circuit pictured to the right to answer questions (a), (b), and (c). a.
What is the total resistance of the circuit?
b.
What is the current in the circuit?
c.
What is the voltage drop across each resistor?
Use the series circuit pictured to the right to answer questions (a)-(e). a.
What is the total voltage of the circuit?
b.
What is the total resistance of the circuit?
c.
What is the current in the circuit?
d.
What is the voltage drop across each light bulb?
e.
Draw the path of the current on the diagram.
Use the series circuit pictured right to answer questions (a), (b), and (c). Consider each resistor equal to all others. a.
What is the resistance of each resistor?
b.
What is the voltage drop across each resistor?
c.
On the diagram, show the amount of voltage in the circuit before and after each resistor.
Use the series circuit pictured right to answer questions (a) (d). a.
What is the total resistance of the circuit?
b.
What is the current in the circuit?
c.
What is the voltage drop across each resistor?
d.
What is the sum of the voltage drops across the three resistors? What do you notice about this sum?
Use the diagram to the right to answer questions (a), (b), and (c). a.
How much current would be measured in each circuit if each light bulb has a resistance of 6 ohms?
b.
How much current would be measured in each circuit if each light bulb has a resistance of 12 ohms?
c.
What happens to the amount of current in a series circuit as the number of batteries increases?
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20.1C Parallel Circuits A parallel circuit has at least one point where the circuit divides, creating more than one path for current. Each path is called a branch. The current through a branch is called branch current. If current flows into a branch in a circuit, the same amount of current must flow out again, This rule is known as Kirchoff’s current law. Because each branch in a parallel circuit has its own path to the battery, the voltage across each branch is equal to the battery’s voltage. If you know the resistance and voltage of a branch you can calculate the current with Ohm’s Law (I=V/R).
1.
2.
3.
4.
Use the parallel circuit pictured right to answer questions (a) - (d). a.
What is the voltage across each bulb?
b.
What is the current in each branch?
c.
What is the total current provided by the battery?
d.
Use the total current and the total voltage to calculate the total resistance of the circuit.
Use the parallel circuit pictured right to answer questions (a) - (d). a.
What is the voltage across each bulb?
b.
What is the current in each branch?
c.
What is the total current provided by the battery?
d.
Use the total current and the total voltage to calculate the total resistance of the circuit.
Use the parallel circuit pictured right to answer questions (a) - (d). a.
What is the voltage across each resistor?
b.
What is the current in each branch?
c.
What is the total current provided by the batteries?
d.
Use the total current and the total voltage to calculate the total resistance of the circuit.
Use the parallel circuit pictured right to answer questions (a) - (c). a.
What is the voltage across each resistor?
b.
What is the current in each branch?
c.
What is the total current provided by the battery?
20.1C Parallel Circuits
In part (d) of problems 1, 2, and 3, you calculated the total resistance of each circuit. This required you to first find the current in each branch. Then you found the total current and used Ohm’s law to calculate the total resistance. Another way to find the total resistance of two parallel resistors is to use the formula shown below.
R to ta l
R1 u R 2 R1 R 2
Calculate the total resistance of a circuit containing two 6 ohm resistors. Given The circuit contains two 6 : resistors in parallel. Looking for The total resistance of the circuit. Relationships R to ta l
1.
2.
Solution
R to ta l R to ta l
6:u6: 6: 6: 3:
The total resistance is 3 ohms.
R1 u R 2 R1 R 2
Calculate the total resistance of a circuit containing each of the following combinations of resistors. a.
Two 8 : resistors in parallel
b.
Two 12 : resistors in parallel
c.
A 4 : resistor and an 8 : resistor in parallel
d.
A 12 : resistor and a 3 : resistor in parallel
To find the total resistance of three resistors A, B, and C in parallel, first use the formula to find the total of resistors A and B. Then use the formula again to combine resistor C with the total of A and B. Use this method to find the total resistance of a circuit containing each of the following combinations of resistors a.
Three 8 : resistors in parallel
b.
Two 6 : resistors and a 2 : resistor in parallel
c.
A 1 :, a 2 :, and a 3 : resistor in parallel
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