We can now substitute what we know into the equation to solve for t.
Remember that this is only the trip up though. To solve for the total time the ball in the air we simply double the answer.
c) We know the ceiling height , the initial velocity (), and the acceleration (). Using the equation , we can solve for v.
We can now find the velocity.
Example 3
Question: Two cars are heading toward each other, traveling at (car A) and (car B). They are apart. How much time do they have before they collide?
Answer: To find the answer, we must find the proportion of the distance that each car travels. In the time that car A travels , car B will travel . This can be made into the ratio which can then be simplified into . This means that car A travels (which is ) of the distance and car B travels (which is ) of the distance. Car B takes to cover . Car A takes to cover . Therefore the time before the collision is 6 minutes.
One-Dimensional Motion Problem Set
Answer the following questions about one-dimensional motion. What is the difference between distance and displacement Write a few sentences explaining this.
Does the odometer reading in a car measure distance or displacement?
Imagine a fox darting around in the woods for several hours. Can the displacement of the fox from his initial position ever be larger than the total distance d he traveled? Explain.
What is the difference between acceleration and velocity? Write a paragraph that would make sense to a grader.
Give an example of a situation where an object has an upward velocity, but a downward acceleration.
What is the difference between average and instantaneous velocity? Make up an example involving a trip in a car that demonstrates your point.
If the position of an object is increasing linearly with time (i.e., is proportional to ), what can we say about its acceleration? Explain your thinking.
If the position of an object is increasing non-linearly with time (i.e., is not proportional to ), what can we say about its velocity? Explain your thinking.
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. Your speed and his speed are the same.
Your position along the highway is the same as his position along the highway.
Your acceleration and his acceleration are the same.
If a car is slowing down from to , but the position is increasing, which of the following statements is true? You may choose more than one. The velocity of the car is in the direction.
The acceleration of the car is in the same direction as the velocity.
The acceleration of the car is in the opposite direction of the velocity.
The acceleration of the car is in the direction.
A horse is galloping forward with an acceleration of . Which of the following statements is necessarily true? You may choose more than one. The horse is increasing its speed by every second, from to to to .
The speed of the horse will triple every second, from to to to .
Starting from rest, the horse will cover of ground in the first second.
Starting from rest, the horse will cover of ground in the first second.
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 . Each car image to the right represents times , and seconds later.
Who is ahead at ? Explain.
Who is accelerating? Explain.
Who is going fastest at ? Explain.
Which car has a constant velocity throughout? Explain.
Graph vs. and vs. . Put both cars on same graph; label which line is which car.
Which car is going faster at (Hint: Assume they travel the same distance between and seconds)?
In the picture below, 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 your graphs with the times indicated in the picture.
Draw the position vs. time graph that corresponds to the velocity vs. time graph below. You may assume a starting position . Label both axes of your graph with appropriate values.
Two cars are heading right towards each other, but are apart. One car is going and the other is going . How much time do they have before they collide head on?
The following data represent the first 30 seconds of actor Crispin Glover’s drive to work. {
Sketchy LeBaron, a used car salesman, claims his car is able to go from to in seconds. What is the average acceleration of this car? Give your answer in . (Hint: you will have to perform a conversion.)
How much distance does this car cover in these seconds? Express your answer twice: in meters and in feet.
What is the speed of the car in after seconds?
Michael Jordan had a vertical jump of about inches. Convert this height into meters.
Assuming no air resistance, at what speed did he leave the ground?
What is his speed of the way up?
What is his speed just before he hits the ground on the way down?
You are sitting on your bike at rest. Your brother comes running at you from behind at a speed of . At the exact moment he passes you, you start up on your bike with an acceleration of . Draw a picture of the situation, defining the starting positions, speeds, etc.
At what time do you have the same speed as your brother?
At what time do you pass your brother?
Draw another picture of the exact moment you catch your brother. Label the drawing with the positions and speeds at that moment.
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.)
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.)
You are standing at the foot of the Bank of America building in San Francisco, which is floors high. You launch a ball straight up in the air from the edge of the foot of the building. The initial vertical speed is . (For this problem, you may ignore your own height, which is very small compared to the height of the building.) How high up does the ball go?
How fast is the ball going right before it hits the top of the building?
For how many seconds total is the ball in the air?
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 that of Earth. Assume you launch upwards with the same speed on the Moon as you do on the Earth.
A car is smashed into a wall during Weaverville’s July Destruction Derby. The car is going just before it strikes the wall. It comes to a stop seconds later. What is the average acceleration of the car during the collision?
A helicopter is traveling with a velocity of 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 above the pillow, he lets go of the rope. What is the speed of the man just before he lands on the pillow?
How long is he in the air after he lets go?
What is the greatest height reached by the man above the ground? (Hint: this should be greater than . Why?)
What is the distance between the helicopter and the man three seconds after he lets go of the rope?
You are speeding towards a brick wall at a speed of . The brick wall is only feet away. What is your speed in ?
What is the distance to the wall in meters?
What is the minimum acceleration you should use to avoid hitting the wall?
What acceleration should you use to increase your speed from to over a distance of ?
You
drop a rock from the top of a cliff. The rock takes seconds to reach the bottom. What is the initial speed of the rock?
What is the magnitude (i.e., numerical value) of the acceleration of the rock at the moment it is dropped?
What is the magnitude of the acceleration of the rock when it is half-way down the cliff?
What is the height of the cliff?
An owl is flying along above your farm with positions and velocities given by the formulas
What is the acceleration of the owl?
What is the speed of the owl at ?
Fill in the missing elements of the table.
{
For each of the following graphs, write a few sentences about what kind of motions were made. Try to use the words we have defined in class (speed, velocity, position, acceleration) in your description.
Answers to Selected Problems
.
.
.
.
a. Zyan b. Ashaan is accelerating because the distance he travels every 0.1 seconds is increasing, so the speed must be increasing
c. Ashaan
d. Zyan
f. Ashaan
.
.
6 minutes
d. e.
f.
g.
h. Between and sec because your position goes from to .
i. You made some sort of turn
a. b.
c.
a. b.
c.
d.
b. 1 second c. at 2 seconds d.
a. b.
c. for round trip
Let’s say we can jump in the air. ? Then, on the moon, we can jump straight up.
b. 3.6 seconds
c. d.
b.
c.
a. b.
c.
d.
a. b.
Chapter 5: Two-Dimensional and Projectile Motion Version 2
The Big Idea
In this chapter, we explore the motion of projectiles under the influence of gravity --- fired cannonballs, thrown basketballs, and other objects that have no way of propelling themselves and do not experience significant air resistance. From chapter 1, we know that vectors can be separated into components; if they are separated into perpendicular components the motion along each component can be treated independently (figure 1).
This is the insight that allows us to solve two dimensional projectile motion problems: we break any initial velocity vector into a component parallel to the ground and a component perpendicular to it. The force of gravity --- which will be explained in more detail later --- accelerates any object near the surface of the earth toward its center at a rate of . This acceleration is in the direction perpendicular to the surface of the earth, conventionally labeled .
Since in projectile motion under the sole influence of gravity any acceleration the object experiences is in the direction, its horizontal, or , velocity remains constant throughout its flight (at least in the absence of air resistance, which we ignore for the time being). To solve two dimensional motion problems, we apply the kinematics equations of one-dimensional motion to each of the two directions. In the direction, we can use the uniform acceleration equations to solve for time in flight. Using this time, we can find how far the object traveled in the direction also.
Solving Two Dimensional Motion Problems
Break the Initial Velocity into its Components
Apply the Kinematics Equations
Key Concepts
In projectile motion, the horizontal displacement of an object from its starting point is called its range.
Vertical () speed is zero only at the highest point of a thrown object's flight.
To work these problems, separate the “Big Three” equations into two sets: one for the vertical direction, and one for the horizontal. Keep them separate.
The only variable that can go into both sets of equations is time. You use time to communicate between the two directions.
Since in the absence of air resistance there is no acceleration in the horizontal direction, this component of velocity does not change over time. This is a counter-intuitive notion for many. (Air resistance will cause velocity to decrease slightly or significantly depending on the object. But this factor is ignored for the time being.)
Motion in the vertical direction must include the acceleration due to gravity, and therefore the velocity in the vertical direction changes over time.
The shape of the path of an object undergoing projectile motion in two dimensions is a parabola.
Two Dimensional Example
Example 1
Question: A ball of mass is moving horizontally with a speed of off a cliff of height . How much time does it take the ball to travel from the edge of the cliff to the ground? Express your answer in terms of (acceleration due to gravity) and (height of the cliff).
Solution: Since we are solving for the time, any motion in the direction is not pertinent. We can just use the equation and solve for t. Notice though that , the ball's initial velocity in the direction, is equal to zero when the ball rolls of the cliff. We can therefore disregard it; we have:
Though we have solved for , we have not solved for it in terms of the given quantities We can replace with because the only acceleration on the ball is due to gravity. We now need to replace with some combination of and . Using the equation we can solve for in the terms wanted. Note that here denotes the distance traveled by the object, or . It isn't the horizontal . Because , we can once again disregard it. We now replace with and with . This gives us Because we have solved for we will make the other equation --- the one solved for --- also contain . Specifically, Now we can substitute for . We can then solve for the answer:
Two Dimensional Motion Problem Set
Draw detailed pictures for each problem (putting in all the data, such as initial velocity, time, etc.), and write down your questions when you get stuck.
Determine which of the following is in projectile motion. Remember that “projectile motion” means that gravity is the only means of acceleration for the object. A jet airplane during takeoff.
A baseball during a Barry Bonds home run.
A spacecraft just after all the rockets turn off in Earth orbit.
A basketball thrown towards a basket.
A bullet shot out of a gun.
An inter-continental ballistic missile.
A package dropped out of an airplane as it ascends upward with constant speed.
Decide if each of the statements below is True or False. Then, explain your reasoning. At a projectile’s highest point, its velocity is zero.
At a projectile’s highest point, its acceleration is zero.
The rate of change of the position is changing with time along the projectile path.
The rate of change of the position is changing with time along the projectile path.
Suppose that after , an object has traveled in the horizontal direction. If the object is in projectile motion, it must travel in the vertical direction as well.
Suppose a hunter fires his gun. Suppose as well that as the bullet flies out horizontally and undergoes projectile motion, the shell for the bullet falls directly downward. Then, the shell hits the ground before the bullet.
Imagine the path of a soccer ball in projectile motion. Which of the following is true at the highest point in its flight? .
.
.
.
.
A hunter with an air blaster gun is preparing to shoot at a monkey hanging from a tree. He is pointing his gun directly at the monkey. The monkey’s got to think quickly! What is the monkey’s best chance to avoid being smacked by the rubber ball? The monkey should stay right where he is: the bullet will pass beneath him due to gravity.
The monkey should let go when the hunter fires. Since the gun is pointing right at him, he can avoid getting hit by falling to the ground.
The monkey should stay right where he is: the bullet wil
l sail above him since its vertical velocity increases by every second of flight.
The monkey should let go when the hunter fires. He will fall faster than the bullet due to his greater mass, and it will fly over his head.
You are riding your bike in a straight line with a speed of . You accidentally drop your calculator out of your backpack from a height of above the ground. When it hits the ground, where is the calculator in relation to the position of your backpack? (Neglect air resistance.) You and your backpack are ahead of the calculator.
You and your backpack are directly above the calculator.
You and your backpack are behind the calculator.
None of the above.
A ball of mass is moving horizontally with speed off a cliff of height , as shown. How much time does it take the rock to travel from the edge of the cliff to the ground? .
CK-12 People's Physics Book Version 2 Page 3