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The physics instructor who performed the experiment in problem 46 changed the procedure. The mass was varied while the force was kept constant. The acceleration of each mass was recorded.
Write the equation relating acceleration to mass given by the data in the graph. Find the units of the constant in the equation. Critical Thinking Problems Find the approximate time needed for a pitched baseball to reach home plate. Report your result to one significant digit.
Use a reference source to find the distance thrown and the speed of a fastball. Answers will vary with the data available.
Have a student walk across the front of the classroom. Estimate his or her walking speed. Estimates will vary. According to the graph, what is the relationship between mass and the acceleration produced by a constant force? How high can you throw a ball? Find a tall building whose height you can estimate and compare the height of your throw to that of the building. Use a graphing calculator or computer graphing program to graph reaction and braking distances versus original speed.
Use the calculator or computer to find the slope of the reaction distance and the best quadratic fit to the braking distance. If the sun suddenly ceased to shine, how long would it take Earth to become dark? You will have to look up the speed of light in a vacuum and the distance from the sun to Earth. How long would it take to become dark on the surface of Jupiter? A student drops a ball from a window 3. The ball accelerates at 9. How fast is it moving when it hits the sidewalk?
A bike travels at a constant speed of 4. How far does it go? A bike accelerates from 0. What distance does it travel? A bike first accelerates from 0. What is the total distance traveled by the bike? He takes 0. How far does the car go before it stops? If you throw the ball in problem 22 up instead of down, how fast is it moving when it hits the sidewalk? Hint: Its acceleration is the same whether it is moving up or down. You throw a ball downward from a window at a speed of 2.
How fast is it moving when it hits the sidewalk 2. Critical Thinking Problems Each of the following problems involves two objects. Draw the pictorial and physical models for each. Use different symbols to represent the position, velocity, and acceleration of each object. Do not solve the problem. A truck is stopped at a stoplight.
When the light turns green, it accelerates at 2. Where and when does the truck catch up with the car? The lowercase symbols represent the cars position, velocity, and accleration. The uppercase symbols represent the trucks position, velocity, and acceleration. Where do they meet? A car is driven km due west, then 65 km due south. What is the magnitude of its displacement? What is the magnitude of your displacement when you follow directions that tell you to walk m in one direction, make a 90 turn to the left and walk m, then make a 30 turn to the right and walk m?
A shopper walks from the door of the mall to her car m down a lane of cars, then turns 90 to the right and walks an additional 60 m.
What is the magnitude of the displacement of her car from the mall door? A hiker walks 4. What is the magnitude of her displacement? What are the magnitude and direction of the change in velocity? What is the planes speed with respect to the ground? You are riding in a bus moving slowly through heavy traffic at 2. You hurry to the front of the bus at 4. What is your speed relative to the street? What is the velocity of the motorboat with respect to the shore?
What are the components of a vector of magnitude 1. A boat is rowed directly upriver at a speed of 2. Viewers on the shore find that it is moving at only 0. What is the speed of the river? Is it moving with or against the boat? A hiker walks Find the east and north components of this walk. What is the velocity both magnitude and direction of the motorboat with respect to the shore?
What are the east and north components of the planes velocity? A golf ball, hit from the tee, travels m in a direction 25 south of the east axis. What are the east and north components of its displacement? What are the planes speed and direction with respect to the ground?
A car moves 65 km due east, then 45 km due west. What is its total displacement? In which direction should the plane head in order to land at an airport due north of its present location? What would be the planes speed with respect to the ground? Graphically find the sum of the following pairs of vectors whose lengths and directions are shown in Figure Tailwind is in the same direction as the airplane Head wind is in the opposite direction of the airplane.
An airplane flies at What is the velocity of the plane relative to the ground if it flies with. Path A is 8. Path B is 7. Path C is 4. Graphically add the hikers displacements in the order A, B, C. N B 35 What can you conclude about the resulting displacements?
You can add vectors in any order. The result is always the same. A river flows toward the east. Because of your knowledge of physics, you head your boat 53 west of north and have a velocity of 6. What is the velocity of the current? A ship leaves its home port expecting to travel to a port Before it moves even 1 km, a severe storm blows it How far is the ship from its destination?
In what direction must it travel to reach its destination? What is your speed relative to the water? You walk 30 m south and 30 m east. Find the magnitude and direction of the resultant displacement both graphically and algebraically.
A descent vehicle landing on Mars has a vertical velocity toward the surface of Mars of 5. At the same time, it has a horizontal velocity of 3. At what speed does the vehicle move along its descent path? At what angle with the vertical is this path?
At the end of three days, the hiker is lost. By drawing a diagram, compute how far the hiker is from camp and which direction should be taken to get back to camp. Take north and east to be positive directions. To return to camp, the hiker must go 3 km east and 4 km south.
You are piloting a small plane, and you want to reach an airport km due south in 3. A wind is blowing from the west at What heading and airspeed should you choose to reach your destination in time? You row a boat perpendicular to the shore of a river that flows at 3. The velocity of your boat is 4. What is the velocity of your boat relative to the shore? What is the speed of the shell relative to the ground?
A rocket in outer space that is moving at a speed of 1. Hot gases are expelled out the rear at 2. What is the speed of the gases relative to the observer? What are the magnitude and direction of the velocity of the balloon? It then moves forward at 2. It immediately goes backward at a speed of 1. Describe in words the motion of the four walkers shown by the four lines in Figure Assume the positive direction is east and the origin is the corner of High Street.
A starts at High St. B starts west of High St. C walks west from High St. D starts east of High St. Answer the following questions about the car whose motion is graphed in Figure When was the car 20 m west of the origin?
Between 10 and 30 s. A remains stationary. B starts at the origin and moves forward at a constant speed. C starts east positive direction of the origin and moves forward at the same speed as B.
D starts at the origin and moves forward at Build Your Own Electric Boat Motor Worksheet a slower speed than B. Draw a position-time graph for a person who starts on the positive side of the origin and walks with uniform motion toward the origin.
Repeat for a person who starts on the negative side of the origin and walks toward the origin. Use the factor-label method to convert the units of the following average velocities. Use data from the graph of the airplanes motion in Figure 56 to convince Chris that this is not true. Consider the motion of bike rider C in Figure Write the equation that represents her motion.
When will rider C be at Draw a position-time graph of a person who walks one block at a moderate speed, waits a short time for a traffic light, walks the next block slowly, and then walks the final block quickly. All blocks are of equal length. A car starts 2. Choose a coordinate system in which the x-axis points east and the origin is at the town square. Write the equation that represents the motion of the car. When will the car reach the town square? At the same time the car in problem 11 left, a truck was 4.
Use the same coordinate system as you did for problem Draw a graph showing the motion of both the car and the truck. Find the time and place where the car passed the truck using both the graph and your two equations. They pass each other when 2. Sketch the velocity-time graphs for the three bike riders in Figure The car runs out of gas, so the driver walks in the same direction at 1.
After spending Use Figure to determine the velocity of the airplane that is speeding up at. Draw a velocity-time graph for the driver, using seconds as your time unit.
You will have to calculate the distance the driver walked to the gas station in order to find the time it took the driver to walk back to the car. Draw a position-time graph for the problem using the areas under the curve of the velocity-time graph. If the bus took twice as long to stop, how would the acceleration compare with what you found in part a? Look at the v -t graph of the toy train in Figure What is its average acceleration?
What is the average acceleration of the bus while braking? A car is coasting downhill at a speed of 3. After 2. Assuming that uphill is the positive direction, what is the cars average acceleration? During which time interval or intervals is the speed constant? During which interval or intervals is the trains acceleration positive? During which time interval is its acceleration most negative? Using Figure , find the average acceleration during the following time intervals. A golf ball rolls up a hill toward a miniature-golf hole.
Assign the direction toward the hole as being positive. If the ball starts with a speed of 2. If the constant acceleration continues for 6. If a car accelerates from rest at a constant 5. How many seconds are required before the car is traveling at 3. The balls velocity simply decreased in the first case. In the second case, the ball slowed to a stop and then began rolling back down the hill.
How far does it move during this time? A bus, traveling at What velocity does it reach 6. For all problems, sketch the situation, assign variables, create a motion diagram, and then develop a mathematical model. How long does it take to achieve this speed? A bike rider accelerates constantly to a velocity of 7. The bikes displacement during the acceleration was 19 m.
What was the initial velocity of the bike? An airplane starts from rest and accelerates at a constant 3. A brick is dropped from a high scaffold. What is its velocity after 4.
How far does the brick fall during this time? A tennis ball is thrown straight up with an initial speed of It is caught at the same distance above ground.
How high does the ball rise? How long does the ball remain in the air? Hint: The time to rise equals the time to fall. Can you show this? You and a friend each drive You travel at How long will your friend wait for you at the end of the trip?
So, it is in the air for 4. A spaceship far from any star or planet accelerates uniformly from How far does it move? Draw a position-time graph of the motion of the ball.
When setting up the axes, use five divisions for each 10 m of travel on the d-axis. Use five divisions for 1 s of time on t-axis. Light from the sun reaches Earth in 8. The velocity of light is 3. How far is Earth from the sun? The total distance a steel ball rolls down an incline at various times is given in Table What distance has the ball rolled at the end of 2.
Plot a position-time graph of the cyclists location from point A at Plot the data in Table Build Your Own Outboard Boat Motor 12 54 on a positiontime graph. Find the average velocity in the time interval between 0.
From the position-time graph in Figure , construct a table showing the average velocity of the object during each s interval over the entire s. You drive a car for 2. What is your average velocity? Do you get the same answer if you drive 1. Total distance is 2. Draw a position-time graph showing the motion of both cars.
How far are the two cars from school when the clock reads 2. Calculate the distances using the equation for motion and show them on your graph. Total time is 4. Do this problem on a worksheet. Both car A and car B leave school when a clock reads zero.
Both cars passed a gas station km from the school. When did each car pass the gas station? Calculate the times and show them on your graph. Car B starts from school at P. When does each car get to the beach? Two cars travel along a straight road.
Answer the following questions, first, graphically by creating a position-time graph, and second, algebraically by writing down equations for the positions dA and dB as a function of the stopwatch time, t. When the cars pass, how long will it have been since car A was at the reference point? A child suddenly runs into the street. If it takes the driver 0. Find the instantaneous speed of the car in Figure at 15 s.
Student answers will vary. You ride your bike for 1. Average velocity is the total displacement divided by the total time. How does the average acceleration of the object in the interval between 0 s and 2 s differ from the average acceleration in the interval between 7 s and 12 s?
Plot a velocity-time graph using the information in Table 55, then answer the questions. TABLE What is its final velocity? Determine the final velocity of a proton that has an initial velocity of 2. How far has it traveled after 6. How far has it gone after 9. How many times the speed of sound is the planes final speed? A plane travels 5. What final velocity does it attain?
How many meters will it take to stop a car going twice as fast? These planes are capable of being accelerated uniformly at the rate of 2. How long will it take the planes to reach takeoff speed? Engineers are developing new types of guns that might someday be used to launch satellites as if they were bullets. One such gun can give a small object a velocity of 3. What acceleration does the gun give this object? Over what time interval does the acceleration take place? Highway safety engineers build soft barriers so that cars hitting them will slow down at a safe rate.
A person wearing a seat belt can withstand an acceleration of 3. What must be the minimum length of the runway? The acceleration occurs as the pitcher holds the ball in his hand and moves it through an almost straightline distance of 3.
Calculate the acceleration, assuming it is uniform. Compare this acceleration to the acceleration due to gravity, 9. Rocket-powered sleds are used to test the responses of humans to acceleration. Calculate the acceleration of the sled when starting, and compare it to the magnitude of the acceleration due to gravity, 9.
Find the acceleration of the sled when braking and compare it to the magnitude of the acceleration due to gravity. The velocity of an automobile changes over an 8. Plot the velocity-time graph of the motion. Determine the displacement of the car during the first 2. What displacement does the car have during the first 4. What does this slope represent?
Figure shows the position-time and velocity-time graphs of a karate experts fist as it breaks a wooden board. Determine the area under the velocitytime curve to find the displacement of the fist in the first 6 ms. Compare this with the position-time graph. The driver of a car going It takes the driver 0.
Determine whether the car hits the barrier. What does this slope indicate? It then suddenly comes to a halt accelerates. Estimate the slope of the velocity-time graph to determine the acceleration of the fist when it suddenly stops. What displacement does the car have during the entire 8. Use the velocity-time graph to describe the motion of the experts fist during the first 10 ms.
Plot the braking distance versus the initial velocity. Describe the shape of the curve. What is the maximum speed at which the car could be moving and not hit the barrier Assume that the acceleration rate doesnt change. Plot the braking distance versus the square of the initial velocity. Calculate the slope of your graph from part b. What is the value of a? The data in Table 57, taken from a drivers handbook, show the distance a car travels when it brakes to a halt from a specific initial velocity.
As a traffic light turns green, a waiting car starts with a constant acceleration of 6. How far will the car travel before it overtakes the truck? Do the graphs confirm the answer you calculated for problem 64? Yes; The graphs confirm the calculated answer. How fast will the car be traveling when it overtakes the truck? Use the information given in problem Draw velocity-time and position-time graphs for the car and truck.
An astronaut drops a feather from 1. If the acceleration of gravity on the moon is 1. What is the stones displacement during this time? A student drops a penny from the top of a tower and decides that she will establish a coordinate system in which the direction of the pennys motion is positive.
What is the sign of the acceleration of the penny? During a baseball game, a batter hits a high pop-up. If the ball remains in the air for 6. Hint: Calculate the height using the second half of the trajectory. The time to fall is 3. The direction of the velocity is positive, and velocity is increasing. Therefore, the acceleration is also positive. A bag is dropped from a hovering helicopter. When the bag has fallen 2. A weather balloon is floating at a constant height above Earth when it releases a pack of instruments.
If the pack hits the ground with a velocity of How long did it take for the pack to fall? Use the data to plot a velocity-time graph. Use the data in the table to plot a position-time graph. Table 58 gives the positions and velocities of a ball at the end of each second for the first 5. Find the slope of the curve at the end of 2. Do the values agree with the table of velocity? Use the data in the table to plot a position-versus-time-squared graph. What type of curve is obtained?
The helicopter in problems 69 and 73 now descends at 5. The same helicopter as in problem 69 is rising at 5. Find the slope of the line at any point. Explain the significance of the value. The helicopter has fallen 1.
What is common to the answers to problems 69, 73, and 74? The bag is 2. A tennis ball is dropped from 1. It rebounds to a height of 1.
With what velocity does it hit the ground? With what velocity does it leave the ground? An express train, traveling at The express engineer spots a local train exactly 1. The local engineer is unaware of the situation.
The express engineer jams on the brakes and slows the express 2 at a constant rate of 3. If the speed of the local train is To solve this problem, take the position of the express train when it first sights the local train as a point of origin. Next, keeping in mind that the local train has exactly 2 a 1.
On the basis of your calculations, would you conclude that a collision will Build Your Own Outboard Boat Motor 500 occur? If the tennis ball were in contact with the ground for 0. Compare the acceleration to g. The calculations you made do not allow for the possibility that a collision might take place before the end of the 12 s required for the express train to come to a halt. To check this, take the position of the express train when it first sights the local train as the point of origin and calculate the position of each train at the end of each second after sighting.
Make a table showing the distance of each train from the origin at the end of each second. Plot these positions on the same graph and draw two lines. Use your graph to check your answer to part a.
What must your average speed be in the second half of the trip to meet your goal? Is this reasonable? Note that the velocities are based on half the distance, not half the time. Draw pictorial models for the following situations. Circle each system. Draw the forces exerted on the system. Name the agent for each force acting on each system. Two horizontal forces, N and N, are exerted in the same direction on a crate. Find the net horizontal force on the crate.
If the same two forces are exerted in opposite directions, what is the net horizontal force on the crate? Be sure to indicate the direction of the net force. The N force is exerted on the crate toward the north and the N force is exerted toward the east. Find the magnitude and direction of the net force. Your hand exerts a 6. Considering the force of gravity on the sugar, what is the net force on the sugar? Give the magnitude and direction.
The downward force is one pound, or 4. The force is. Is the force the same if you lie on the floor? For each problem, draw a motion diagram and a free-body diagram labeling all forces with their agents and indicating the direction of the acceleration and the net force. Draw arrows the appropriate lengths. A skydiver falls downward through the air at constant velocity air drag is important.
Air drag on diver. A rocket blasts off and its vertical velocity increases with time ignore air drag. A cable pulls a crate at constant speed across a horizontal surface there is friction. On Earth, a scale shows that you weigh N. What is your mass? Scale reads N. Since there is no acceleration your force equals the downward force of gravity.
On the moon the scale would read Use the results from the first example problem to answer these questions about a scale in an elevator on Earth. What force would the scale exert when a.
A boy exerts a N horizontal force as he pulls a N sled across a cement sidewalk at constant speed. What is the coefficient of kinetic friction between Build Your Own Outboard Boat Motor Job Description the sidewalk and the metal sled runners? Ignore air resistance. Suppose the sled runs on packed snow. The coefficient of friction is now only 0. If a person weighing N sits on the sled, what force is needed to pull the sled across the snow at constant speed?
Consider the doubled force pushing the crate in the example problem Unbalanced Friction Forces. How long would it take for the velocity of the crate to double to 2. The initial velocity is 1. Depends upon the magnitude of the acceleration.
Would it be practical to make a pendulum with a period of Calculate the length and explain. On a planet with an unknown value of g, the period of a 0. What is g for this planet? A kg lb dragster, starting from rest, attains a speed of Find the average acceleration of the dragster during this time interval. What is the magnitude of the average net force on the dragster during this time?
You lift a bowling ball with your hand, accelerating it upward. What are the forces on the ball? What are the other parts of the action-reaction pairs? On what objects are they exerted? The force of your hand on the ball, the gravitational force of Earths mass on the ball.
The force of the ball on your hand, the gravitational force of the balls mass on Earth. The force of your feet on Earth, the force of Earth on your feet. A car brakes to a halt. What forces act on the car?
The backward friction and upward normal force of the road on the tires and the gravitational force of Earths mass on the car. The forward friction and the downward force of the tires on the road and the gravitational force of the cars mass on Earth.
Assume that the driver has a mass of 68 kg. What horizontal force does the seat exert on the driver? The dragster in problem 20 completed the If the car had a constant acceleration, what would be its acceleration and final velocity? After a day of testing race cars, you decide to take your own kg car onto the test track. While moving down the track at What is the average net force that the track has applied to the car during the The swimmer comes to a stop 2.
Find the net force exerted by the water. Section 6. What is your weight in newtons? Your new motorcycle weighs N. What is its mass in kg? The dragster in problem 21 crossed the finish line going Does the assumption of constant acceleration hold true?
What other piece of evidence could you use to see if the acceleration is constant? A race car has a mass of kg. It starts from rest and travels The car is uniformly accelerated during the entire time. What net force is exerted on it? How long would the pendulum have to be to double the period?
Because the period is proportional to the square root of the length, the pendulum would have to be four times as long, or 2. You place a 7. If the scale reads If you use a horizontal force of A kg crate is pushed horizontally with a force of N.
If the coefficient of friction is 0. A kg helicopter accelerates upward at 2. What lift force is exerted by the air on the propellers? The maximum force a grocery sack can withstand and not rip is N. If A force of How large is the frictional force? What is the coefficient of friction? You are driving a As you approach an intersection, the traffic light turns red. You slam on the brakes. Your wheels lock, the tires begin skidding, and the car slides to a halt in a distance of What is the coefficient of kinetic friction between your tires and the icy road?
A student stands on a bathroom scale in an elevator at rest on the 64th floor of a building. The scale reads N. As the elevator moves up, the scale reading increases to N, then decreases back to N. Find the acceleration of the elevator. As the elevator approaches the 74th floor, the scale reading drops to N.
What is the acceleration of the elevator? Once moving, what total force must be applied to the sled to accelerate it at 3. Using your results from parts a and b, explain which change in velocity, starting or stopping, would take the longer time.
The instruments attached to a weather balloon have a mass of 5. The balloon is released and exerts an upward force of 98 N on the instruments. What changes would you expect in the scale readings on the ride back down? When constant downward velocity is reached, the scale reads N because the acceleration is then zero.
When the elevator is slowing at the bottom, the acceleration is positive and the scale reads more than N. What force will be needed to start the sled moving? After the balloon has accelerated for What is the velocity of the instruments at the moment of their release? What net force acts on the instruments after their release?
When does the direction of their velocity first become downward? The velocity becomes negative after it passes through zero. What force is needed to keep the sled moving at a constant velocity?
A sled of mass The static friction coefficient is 0. The hanging masses are free to move. Choose coordinate systems for the two masses with the positive direction up for mA and down for mB. A kg boy and a kg girl use an elastic rope while engaged in a tug-of-war on an icy, frictionless surface. If the acceleration of the girl toward the boy is 3.
As a baseball is being caught, its speed goes from The mass of the baseball is 0. What is the baseballs acceleration? What are the magnitude and direction of the force acting on the player who caught it?
Same magnitude, opposite direction in direction of velocity of ball. Find the acceleration of the smaller mass.
Suppose the masses in problem 41 are now 1. Find the acceleration of the larger mass. Take the direction of the physical motion, smaller mass upward and larger mass downward, to be the positive direction of motion. The force exerted on a 0. Draw a graph of force versus time. What is the average force exerted on the ball by the bat? What is the acceleration of the ball? Fnet 5. What is the final velocity of the ball, assuming that it reverses direction?
What force magnitude and direction is needed to put the weight into equilibrium? The sign from the preceding example problem is now hung by ropes that each make an angle of 42 with the horizontal. What force does each rope exert?
Two ropes pull on a ring. One exerts a N force at What are the magnitude and direction of the resultant force on the weight? Vector addition is most easily carried out by using the method of addition by components.
The first step in this method is the resolution of the given vectors into their horizontal and vertical components. The horizontal and vertical components of the resultant vector are found by simple addition.
The magnitude and direction of the resultant vector are found by the usual method. Two forces are exerted on an object.
A N force acts at and a N force acts at What are the magnitude and direction of the equilibrant? What are the magnitude and direction of the force that would cause the ring to be in equilibrium?
Consider the trunk on the incline in the Example Problem. Calculate the magnitude of the acceleration. After 4. For the Example Problem Skiing Downhill, find the x- and y-components of the weight of the skier going downhill.
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