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Find the missing legs or angles of the triangles shown.
Draw in the and velocity components for each dot along the path of the cannonball. The first one is done for you.
A stone is thrown horizontally at a speed of from the edge of a cliff in height. How far from the base of the cliff will the stone strike the ground?
A toy truck moves off the edge of a table that is high and lands from the base of the table. How much time passed between the moment the car left the table and the moment it hit the floor?
What was the horizontal velocity of the car when it hit the ground?
A hawk in level flight above the ground drops the fish it caught. If the hawk’s horizontal speed is , how far ahead of the drop point will the fish land?
A pistol is fired horizontally toward a target away, but at the same height. The bullet’s velocity is . How long does it take the bullet to get to the target? How far below the target does the bullet hit?
A bird, traveling at , wants to hit a waiter below with his dropping (see image). In order to hit the waiter, the bird must release his dropping some distance before he is directly overhead. What is this distance?
Joe Nedney of the San Francisco 49ers kicked a field goal with an initial velocity of at an angle of . How long is the ball in the air? Hint: you may assume that the ball lands at same height as it starts at.
What are the range and maximum height of the ball?
A racquetball thrown from the ground at an angle of and with a speed of lands exactly later on the top of a nearby building. Calculate the horizontal distance it traveled and the height of the building.
Donovan McNabb throws a football. He throws it with an initial velocity of at an angle of . How much time passes until the ball travels horizontally? What is the height of the ball after seconds? (Assume that, when thrown, the ball is above the ground.)
Pablo Sandoval throws a baseball with a horizontal component of velocity of . After seconds, the ball is above the release point. Calculate the horizontal distance it has traveled by this time, its initial vertical component of velocity, and its initial angle of projection. Also, is the ball on the way up or the way down at this moment in time?
Barry Bonds hits a home run that lands in the stands at an altitude above its starting altitude. Assuming that the ball left the bat at an angle of from the horizontal, calculate how long the ball was in the air.
A golfer can drive a ball with an initial speed of . If the tee and the green are separated by , but are on the same level, at what angle should the ball be driven? (Hint: you should use at some point.)
How long will it take a bullet fired from a cliff at an initial velocity of , at an angle below the horizontal, to reach the ground below?
A diver in Hawaii is jumping off a cliff high, but she notices that there is an outcropping of rocks out at the base. So, she must clear a horizontal distance of during the dive in order to survive. Assuming the diver jumps horizontally, what is his/her minimum push-off speed?
If Monte Ellis can jump high on Earth, how high can he jump on the moon assuming same initial velocity that he had on Earth (where gravity is that of Earth’s gravity)?
James Bond is trying to jump from a helicopter into a speeding Corvette to capture the bad guy. The car is going and the helicopter is flying completely horizontally at . The helicopter is above the car and behind the car. How long must James Bond wait to jump in order to safely make it into the car?
A field goal kicker lines up to kick a yard field goal. He kicks it with an initial velocity of at an angle of . The field goal posts are meters high. Does he make the field goal?
What is the ball’s velocity and direction of motion just as it reaches the field goal post (i.e., after it has traveled in the horizontal direction)?
In a football game a punter kicks the ball a horizontal distance of yards . On TV, they track the hang time, which reads seconds. From this information, calculate the angle and speed at which the ball was kicked. (Note for non-football watchers: the projectile starts and lands at the same height. It goes yards horizontally in a time of seconds)
Answers to Selected Problems
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a. b. degrees
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d. degrees,
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a. b.
below target
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a. . b.
seconds, meters
; on the way up
seconds
a. yes b. @ degrees from horizontal
@ degrees
Chapter 6: Newton's Laws Version 2
The Big Idea
In the previous chapters, we studied the behavior of accelerating objects in one and two dimensions. We did not, however, address the issue of where the acceleration comes from: in other words, why, in certain situations, do the velocities of objects change? It might make sense that a cart moves if I push it, but what about a dropped object: is it accelerating for a different reason, or for the same one? Is there something common to all accelerating objects?
Building on the insights of scientists before him, Isaac Newton created a mathematical analysis of moving and accelerating objects; the rules he discovered are now known as Newton's Laws of Motion. Newton is a legendary figure to physicists, and it's hard to underestimate his influence on the field. Actually, the substance of his Laws had been summarized by scientists before him. Still, the mathematical framework for their interpretation that Newton created was a revolutionary achievement, since it unified the existing knowledge of mechanics in a consistent system and cemented math as the accepted method of interpreting physical phenomena.
Here are Newton's Laws, in modern English:
Newton's First Law
Every body continues in its state of rest, or of uniform motion in a right (straight) line, unless it is compelled to change that state by forces impressed upon it.
Newton's Third Law
To every action there is always opposed an equal reaction: or, the mutual actions of two bodies upon each other are always equal, and directed to contrary parts.
Newton's Second Law
The change of motion is proportional to the motive force impressed; and is made in the direction of the right (straight) line in which that force is impressed.
(Taken from the Principia in modern English, Isaac Newton, University of California Press, 1934).
Newton's Laws Explained
The First Law is about inertia; objects at rest stay at rest unless acted upon and objects in motion continue that motion in a straight line unless acted upon. Prior to Newton and Galileo, the prevailing view on motion was still Aristotle's. According to his theory the natural state of things is at rest; force is required to keep something moving at a constant rate. This made sense to people throughout history because on earth, friction and air resistance slow moving objects. When there is no air resistance (or other sources of friction), a situation approximated in space, Newton's first law is much more evident.
The "motion" Newton mentions in the Second Law is, in his language, the product of the mass and velocity of an object --- we call this quantity momentum --- so the Second Law is actually the famous equation: That is, the acceleration experienced by an object will be proportional to the applied force and inversely proportional to its mass. If there are multiple forces, they can be added as vectors and it is the net force that matters: When the net force on an object is zero, it is said to be in translational equilibrium:
Finally, the Third Law states that you can't push someone or something without being pushed back. This law is somewhat confusing: if to each applied force there is an equal and opposite force, why does anything ever accelerate? The key is that the `equal and opposite' forces act on different objects. If I push a cart, the cart is in turn pushing on me. However, I'm also pushing (and being pushed by) the e
arth, through my feet. Therefore, in the end, the cart and I move in the same direction and the earth moves opposite us. The cart-person system experienced a net force in one direction, while the earth experienced an equal and opposite force. According to Newton's second law, the acceleration objects experience due to applied forces is inversely proportional to their mass; clearly, the earth --- with its gigantic mass --- doesn't move very far compared to the cart and person.
Newton's Laws Example
Question: Tom and Mary are standing on identical skateboards. Tom and Mary push off of each other and travel in opposite directions.
a) If Tom and Mary have identical masses, who travels farther?
b) If Tom has a bigger mass than Mary, who goes farther?
c) If Tom and Mary have identical masses and Tom pushes twice as hard as Mary, who goes farther?
Solution
a) Neither. Both Tom and Mary will travel the same distance. The forced applied to each person is the same (Newton's Third Law). So which cancels to Therefore both people will travel the same distance because the acceleration controls how far someone will travel and Tom and Mary have equal acceleration.
b) Mary will go farther. Again, the same force is applied to both Mary and Tom so Since Tom has the larger mass, his acceleration must be smaller (acceleration and mass are inversely proportional). Finally, because Mary's acceleration is greater, she will travel farther.
c) Neither. Newton's Third Law states that for every action there is an equal and opposite reaction. Therefore if Tom pushes twice as hard as Mary, Mary will essentially be pushing back with the same strength. They will therefore travel the same distance.
What are Forces?
In other words, things tend to stay in the their current state of motion unless some "forces" are "impressed" on them. But where do such forces come from? What are they? Force isn't a real object, but rather a concept used to describe actions. We can think of it as the cause of any kind of "pushing" or "pulling" that an object experiences. As long as we can measure them consistently, forces can be treated like any other physical vector quantity. One way to state a major goal of physics is to find a method for consistently predicting the forces an object will experience under any circumstances, based on the circumstances. At this point, physicists have identified four basic forces that govern the universe:
The strong force: The most powerful of the four forces, it holds nuclei together in atoms --- but has a very short range. It has to overcome the massive electromagnetic repulsion between protons in a nucleus.
The electromagnetic force: Responsible for the behavior of charged particles. Has infinite range.
The weak force: Another nuclear force, responsible for much of the structure of stars and the universe in general. Its range is longer than that of the strong force, but still smaller than an atom.
Gravity: Responsible for the attraction of all masses in the universe. Has infinite range.
All others --- friction, air resistance, and other contact forces; buoyancy; the spring force --- can be reduced to these fundamental forces. The fundamental forces are covered in more detail in later chapters.
Note that this classification does not tell us anything about where these forces come from, or how they are able to act seemingly at a distance, with "no strings attached". Newton himself said:
I have not as yet been able to discover the reason for these properties of gravity from phenomena, and I do not feign hypotheses.
In other words, we are interested in describing and predicting nature, rather than explaining its root causes. Forces acting a distance may seem strange in light of our experience with forces that we can apply to things (like pushing on rocks, etc), but to Newton asking about the nature of gravity was like asking about the nature of mass: it's just there, we can measure it, and that's it.
Common Forces
Universal Gravity
In previous chapters we learned that gravity --- near the surface of planets, at least --- is a force that accelerates objects at a constant rate. At this point we can extend this description using the framework of Newton's Laws.
Newton's Laws apply to all forces; but when he developed them only one was known: gravity. Newton's major insight --- and one of the greatest in the history of science --- was that the same force that causes objects to fall when released is also responsible for keeping the planets in orbit. According to some sources, he realized this while taking a stroll through some gardens and witnessing a falling apple.
After considering the implications of this unification, Newton formulated the Law of Universal Gravitation: Any two objects in the universe, with masses and with their centers of mass at a distance apart will experience a force of mutual attraction along the line joining their centers of mass equal to: Here is an illustration of this law for two objects, for instance the earth and the sun:
Gravity on the Earth's Surface
In the chapter on energy, we saw that the gravitational potential energy formula for objects near earth, , is a special case of a more general result. It so happens that the fact that gravity accelerates near earth objects at a constant rate is an almost identical result.
On the surface of a planet --- such as earth --- the in formula [3] is very close to the radius of the planet, since a planet's center of mass is --- usually --- at its center. It also does not vary by much: for instance, the earth's radius is about 6,000 km, while the heights we consider for this book are on the order of at most a few kilometers --- so we can say that for objects near the surface of the earth, the in formula [3] is constant and equal to the earth's radius. This allows us to say that gravity is more or less constant on the surface of the earth. Here's an illustration:
For any object a height above the surface of the earth, the force of gravity may be expressed as: We can do this because the quantity in braces only has constants; we can combine them and call their product . Remember, this is an approximation that holds only when the in formula [3] is more or less constant.
We call the quantity an object's weight. Weight is different from mass --- which is identical everywhere --- since it depends on the gravitational force an object experiences. In fact, weight is the magnitude of that force. To find the weight of an object on another planet, star, or moon, use the appropriate values in formula [4].
Normal Force
Often, objects experience gravitational attraction but cannot move closer together because they are in contact. For instance, when you stand on the surface of the earth you are obviously not accelerating toward its center. According to Newton's Laws, there must be a force opposing gravity, so that the net force on both objects is zero. We call such a force the Normal Force. It is actually electromagnetic in nature (like other contact forces), and arises due to the repulsion of atoms in the two objects. Here is an illustration of the Normal force on a block sitting on earth:
Gravity and Normal Force Example
Question: A woman of mass 70.0 kg weighs herself in an elevator.
c) When the elevator is accelerating upward at 2.00 m/s2, what does the scale read?
b) When the elevator is not accelerating, what does the scale read (i.e., what is the normal force that the scale exerts on the woman)?
a) If she wants to weigh less, should she weigh herself when accelerating upward or downward?
Answer a) If she wants to weigh less, she has to decrease her force (her weight is the force) on the scale. We will use the equation to determine in which situation she exerts less force on the scale.
If the elevator is accelerating upward then the acceleration would be greater. She would be pushed toward the floor of the elevator making her weight increase. Therefore, she should weigh herself when the elevator is going down.
b) When the elevator is not accelerating, the scale would read .
c) If the elevator was accelerating upward at a speed of , then the scale would read which is .
Tension
Another force that often opposes gravity i
s known as tension. This force is provided by wires and strings when they hold objects above the earth. Like the Normal Force, it is electromagnetic in nature and arises due to the intermolecular bonds in the wire or string:
If the object is in equilibrium, tension must be equal in magnitude and opposite in direction to gravity. This force transfers the gravity acting on the object to whatever the wire or string is attached to; in the end it is usually a Normal Force --- between the earth and whatever the wire is attached to --- that ends up balancing out the force of gravity on the object.
Friction
Friction is a force that opposes motion. Any two objects in contact have what is called a mutual coefficient of friction. To find the force of friction between them, we multiply the normal force by this coefficient. Like the forces above, it arises due to electromagnetic interactions of atoms in two objects. There are actually two coefficients of friction: static and kinetic. Static friction will oppose initial motion of two objects relative to each other. Once the objects are moving, however, kinetic friction will oppose their continuing motion. Kinetic friction is lower than static friction, so it is easier to keep an object in motion than to set it in motion. There are some things about friction that are not very intuitive:
The magnitude of the friction force does not depend on the surface areas in contact.
CK-12 People's Physics Book Version 2 Page 4
