Mechanics
Projectile Motion
|
|
|
· For a projectile, describe the changes in the horizontal and vertical components of its velocity, when air resistance is negligible.
· Explain why a projectile moves equal distances horizontally in equal time intervals, when air resistance is negligible.
· Show an understanding of the independence of vertical and horizontal velocities of a projectile.
· Find the velocity of a projectile launched horizontally when a change in the vertical distance occurs.
· Find the maximum height and range of projectiles launched at an arbitrary angle.
Free Fall and Linear Motion Revisited
Activity and Notes
¹ Fold a dollar bill in half the long way. Draw a line on the fold. Have the person next to you place their index finger and thumb along the line but not touching it. Challenge the person next to you to catch the dollar bill. Repeat several times, observing the results.
0.08 m or 8 cm
All objects, when ignoring air resistance, fall at the
same rate, g = -9.8 m/s2 or 9.8 m/s2 down.
free fall – motion under the influence of gravity only.
acceleration due to gravity (g) – a change in velocity due to the affects of free fall; -9.8 m/s2; can be rounded to 10 m/s2 when estimating; vector; defined as positive ® up and negative ® down.
Vectors can be used to represent the increasing speed of
a ball in free fall.
The distance the ball falls each second increases because
the ball is accelerating. This is shown
by drawing a longer vector arrow for each time interval.
constant velocity (v) – motion in a straight line at a constant speed; vf = vo.
Vectors can also be used to represent a ball rolling
horizontally on a table at a constant velocity.
Each vector arrow is drawn the same length to represent
the constant velocity. The velocity
would remain constant but friction makes it slow down and eventually stop.
Demonstrations and Notes
µ Use the Pasco Scientific apparatus to drop a ball moving at a constant horizontal velocity into a cart. Observe the motion of the ball as it falls focusing on the path followed.
µ Drop a ball while walking at a constant horizontal velocity. Observe the motion of the ball as it falls focusing on the path followed.
The same result is seen when combining the motion of the
ball in free fall with the motion of the ball rolling on the table at a
constant velocity. This is seen when
rolling the ball off of the table. The
ball rolling on the table would continue forever in a straight line if gravity
is ignored. The ball in free fall would
continue to increase its speed if air resistance is ignored.
Since the ball is moving at a constant velocity and in
free fall at the same time, the horizontal and vertical vectors are added
together during equal time intervals.
This is done for each time interval until the ball hits the ground.
The path the ball follows can be seen by connecting the
intermittent resultant vectors.
projectile – any object that moves through space acted on only by gravity.
trajectory – the path followed by a projectile.
µ Drop a ball straight down at the same time one is launched horizontally. Determine which ball will hit the ground first.
Summary Review – Horizontal Projectile Motion
ü A projectile is any object that moves through space acted on only by gravity.
ü Projectile motion is the result of adding the vectors of an object in free fall with the vectors of an object experiencing constant linear motion.
ü The horizontal and vertical components of a projectile launched horizontally are independent of one another.
ü The path a projectile follows is called its trajectory.
Problem Solving Horizontal Projectile Motion
Notes
Since the horizontal and vertical components of a
horizontally launched projectile are independent of one other, the equations
pertaining to linear motion and free fall can be used in problem solving. (As long as we look at them separately.)
|
Horizontal |
Vertical |
|
|
|
|
|
|
When solving problems, the following procedures will be
used.
- Make a sketch of the situation and label all
given and unknown quantities.
- Determine what type of projectile motion is
being observed – horizontal.
- List all of the equations that apply to the problem.
- Determine what variables are known and needed.
- Substitute the given quantities into the
equation using proper units.
- Solve the equation, carrying units throughout
the problem.
- Check your answer to make sure it makes sense.
Summary Review – Problem Solving Horizontal Projectile
Motion
ü The equations used for linear motion and free fall apply to all projectile motion problems.
ü The horizontal and vertical components of all projectiles are solved independently.
ü The Problem Solving Strategy is used when solving problems in physics.
Homework – Problem Solving Horizontal Projectile Motion
|
Horizontal |
Vertical |
|
|
|
|
|
|
1. A stone is thrown horizontally at 15 m/s from the top of a cliff 44 meters high. Find how long it takes the stone to reach the bottom of the cliff, how far from the base of the cliff the stone hits the ground, and the velocity of the stone when it hits the ground.
2. A steel ball rolls with a constant velocity across a tabletop 0.950 meters high. It rolls off and hits the ground 0.352 meters horizontally from the edge of the table. Determine how long it took the ball to hit the ground, the velocity of the ball when rolling on the table, and the velocity of the ball when it lands.
3. A car is traveling too fast in the snow. It slides off the road horizontally and lands in deep snow 43.9 meters below the road and 87.7 meters beyond the edge of the road. Determine how long it took the car to fall and the velocity of the car when it left the road. What is the acceleration of the car after it has fallen 10 meters below the edge of the road?
Demonstrations and Notes
µ Use the Pasco Scientific apparatus to launch a ball straight up while moving in a cart at a constant horizontal velocity. Observe the motion of the ball as it leaves and reenters the cart focusing on the path followed.
µ Throw a ball to someone. Observe the path taken by the ball focusing on the shape of the trajectory.
parabola – shape of the trajectory taken by a projectile launched at an angle.
The ball is still a projectile (any object that moves
through space acted on only by gravity) but it is special because the starting
and stopping heights are the same. This
makes the change in total vertical distance zero.
A projectile launched at an angle would continue in a
straight line at a constant velocity if gravity is ignored. However, gravity makes the projectile
accelerate to Earth. This shows us why a
projectile launched at an angle follows a parabolic trajectory.
range (R) – the horizontal distance a projectile travels when returning to its launched height.
Since the projectile is launched at an angle, it now has
both horizontal and vertical velocities.
The horizontal component of the velocity remains
constant. The vertical component of the
velocity changes as the projectile moves up or down. This is shown when looking at the vector
components of the trajectory at regular time intervals.
The up and right vectors represent the velocity given to
the projectile when launched. The
vertical vectors decrease in magnitude due to gravity. Eventually, the effects of gravity will reduce
the upward velocity to zero. This occurs
at the top of the parabolic trajectory where there is only horizontal motion.
After gravity reduces the upward (vertical) speed to zero
it begins to add a downward velocity.
This velocity increases until the projectile return to the ground.
When looking at each half of the trajectory (up and down)
you can determine that the speed of the projectile going up is equal to the
speed of the projectile coming down.
(Provided air resistance is ignored.)
The only difference is the direction of the motion.
µ Throw the Whistler football straight up. Listen and observe.
µ Use the Pasco Scientific apparatus to Shoot the Monkey.
Summary Review – Angled Projectile Motion
ü Projectiles launched at an angle follow a parabolic trajectory.
ü The range is the horizontal distance a projectile travels when returning to its original height.
ü The horizontal and vertical components of a projectile launched at an angle are independent of one another.
ü The vertical velocity of a projectile launched at an angle when it reaches its maximum height is zero.
ü The speed of a projectile at a given height while traveling up is equal to the speed of the projectile at the same height when traveling down.
Problem Solving Angled Projectile Motion
Notes
Since the horizontal and vertical components of a
projectile launched at any angle are still independent of one other, the
equations pertaining to linear motion and free fall can still be used in
problem solving. (As long as we look at
them separately.)
When solving for time, algebra must be used to reduce the
number of t’s is the equation Recall
that common variables can be factored out of the equation.
and 
The same thing can be done with our equation.
and 
Before the projectile is launched, t = 0, the position is
zero, y = 0. The other half of the
equation is the part that helps us find time; when y at the end of the
trajectory is zero all the time needed to complete the trajectory has passed.
|
Horizontal |
Vertical |
|
|
|
When solving problems, the following procedures will be
used.
- Make a sketch of the situation and label all
given and unknown quantities.
- Determine what type of projectile motion is
being observed – angled.
- List all of the equations that apply to the
problem.
- Determine what variables are known and needed.
- Substitute the given quantities into the
equation using proper units.
- Solve the equation, carrying units throughout
the problem.
- Check your answer to make sure it makes sense.
Changing the angle the projectile is launched affects the
time of flight, maximum height, and range.
Perpendicular angles (any two angles that when added together equal 90°)
produce the same range but varying times and maximum heights.
optimal angle – 45°; achieves maximum range with minimal sacrifice of time and height.
The maximum range of a projectile launched at any angle
is achieved at 45° (exactly half of 90°) or the optimal
angle. Minimal time and height are
sacrificed to obtain a maximum range.
Summary Review – Problem Solving Angled Projectile Motion
ü The equations used for linear motion and free fall apply to all projectile motion problems.
ü The horizontal and vertical components of all projectiles are solved independently.
ü The Problem Solving Strategy is used when solving problems in physics.
Homework – Problem Solving Angled Projectile Motion
|
Horizontal |
Vertical |
|
|
|
1. The punter for the Steelers punts the football with a velocity of 27 m/s at an angle of 30°. Find the ball’s hang time, maximum height, and distance traveled (range) when it hits the ground. (Assume the ball is kicked from ground level.)
2. Repeat the above problem using 60° as the angle the ball is kicked.
3. Repeat the above problem using 45° as the angle the ball is kicked.
4. Compare the results of the above problems. What can be concluded about the relationship between the angle the ball is kicked and the time of flight, maximum height, and range?
Introduction – What’s Going On?
To understand projectile motion, the concepts of linear motion, free fall, and vectors must be intertwined. By combining these concepts, we can predict where a projectile will land whether it rolls off a hill or is fired from the ground. This lab contains two parts that apply the concepts of linear motion, free fall, and horizontal projectile motion in an effort to calculate the landing position of a projectile.
Your job is simply to hit a target with a projectile launched horizontally off the table. You have the choice of working in a group of three or four. To help you remember how to calculate accurate data, a few suggestions are listed below.
|
|
You have done this before! Check your homework and
class notes if you need further
assistance. |
In order to calculate the horizontal distance (x) of a horizontally launched projectile, you need to know the horizontal velocity (vo) of the projectile when it is launched and the time (tfall) the projectile is in flight.
To calculate the horizontal launch velocity (vo) of the projectile, the linear distance (dwood) the ball travels and the time (tavg) it takes to travel that distance must be measured.
To calculate the time (tfall) the projectile is in flight, the vertical height (y) the projectiles falls must be measured. (Remember, g = -9.8 m/s2)
By calculating these values, the horizontal distance (x) the launched projectile travels from the table can be determined.
|
|
|
|
|
|
|
|
|
Percent Error = |
||
Part I – Calculating Horizontal Launch Velocity
- Place the leg of the ramp flat on the table and the wood shelf flat against the end of the ramp. This provides a smooth transition from the ramp to the top of the table. Measure the length of the wood shelf (dwood).
- Place photogate one where the ramp meets the wood shelf. Place photogate two on the outside edge of the wood shelf. Align the photogates so the ball (projectile) can roll between them smoothly. Set the photogates to pulse mode which measures the time it takes the ball to travel between photogates (t1-5).
- Roll the ball down the ramp starting at various locations. (20 cm, 40 cm, 60 cm, 80 cm, and 100 cm) Each location will consist of five time trials.
- Calculate the horizontal launch velocity for each trial using the distance (dwood) the ball travels on the wood shelf and the average time (tavg) it takes to travel that distance.
|
|
DON’T CHEAT!! If the ball hits the floor
before you are ready to Shoot for Your Grade, you will receive a ZERO! |
Part II – Hitting the Target
- Complete the data table to organize your data and keep it in order.
- When you are certain you have accurately calculated the five horizontal distances of the projectile when released at the various positions on the ramp, place the target on the floor and get ready to let the projectiles fly!
Part III – Wrap Up
- The group grade is based on where the projectiles hit the target. (50 possible points)
- Extra credit is awarded for a properly completed data table. (10 possible points)
- No more than 50 total points can be earned.
- To determine the correct horizontal distance (x-corrected) of the projectile (used to calculate percent error), add or subtract the horizontal distance (Dx) you missed the target by to your calculated value. (x-corrected = x ± Dx)
SHOOTING FOR YOUR GRADE LAB II
Introduction – What’s Going On?
To understand projectile motion, the concepts of linear motion, free fall, and vectors must be intertwined. By combining these concepts, we can predict where a projectile will land whether it rolls off a hill or is fired from the ground. This lab contains two parts that apply the concepts of linear motion, free fall, and vectors in an effort to calculate the landing position of a projectile.
Your job is simply to hit a target with a projectile launched at an angle of 45°. You have the choice of working alone (if you don’t think you trust anyone else enough) or in a group of four or less (if you think you may need all the help you can get). To help you remember how to calculate accurate data, a few suggestions are listed below.
In order to calculate the range (R) of a launched projectile, you need to know the horizontal velocity (vo) of the projectile when it is launched and the time (t) the projectile is in flight.
The horizontal and vertical velocities are found using the launch velocity (v) and the angle of launch (q). The time of flight is found using the vertical component of the velocity.
|
|
|
|
|
|
|
||
The launch velocity of the projectile must now be determined. To calculate the launch velocity of the projectile, a different approach is necessary. When firing a projectile horizontally, the horizontal component of the velocity is equal to the actual launch velocity of the projectile. To determine the horizontal velocity of the projectile, simply aim the launcher to fire at 0° over the edge of a table. By measuring the height of the projectile and the horizontal distance it travels when shot the velocity the projectile launcher shoots can be determined.
|
|
|
Part I – Calculating Launch Velocity
- Using the short range setting of the launcher, shoot the projectile from a table horizontally onto the floor. Calculate the launch velocity using the height of the table and horizontal distance the projectile travels.
|
|
You have done this before! Check your homework and
class notes if you need further
help. |
Part II – Hitting the Target
- Using the short range setting of the launcher, PREDICT where the projectile would land if it is shot at an angle q = 45° where the change in vertical position of the projectile is zero. (The launch height and landing heights are equal to produce a parabolic trajectory.)
Part III – Wrap Up
- When you are certain you have accurately calculated the range of the projectile when launched at a 45° angle, place a target on the table with the bull’s-eye center at the range.
|
|
If you DID
NOT hit the A or B range of the target, write a brief
paragraph explaining why you think you missed. It may help
INCREASE your grade! |
|
dwood
= _______ m |
|
|
y =
_______ m |
||
|
TRIAL (ramp distance) |
1 (20 cm) |
2 (40 cm) |
3 (60 cm) |
4 (80 cm) |
5 (100 cm) |
|
t1 (sec) |
|
|
|
|
|
|
t2 (sec) |
|
|
|
|
|
|
t3 (sec) |
|
|
|
|
|
|
t4 (sec) |
|
|
|
|
|
|
t5 (sec) |
|
|
|
|
|
|
tavg (sec) |
|
|
|
|
|
|
|
|
|
|
|
|
|
vo (m/s) |
|
|
|
|
|
|
tfall (sec) |
|
|
|
|
|
|
x (m) |
|
|
|
|
|
|
|
|
|
|
|
|
|
Dx (m) |
|
|
|
|
|
|
x-corrected (m) |
|
|
|
|
|
|
% Error |
|
|
|
|
|
|
Horizontal |
Vertical |
|
|
|
|
|
|
1. A stone is thrown horizontally at 23 m/s from the top of a cliff 51 meters high. How long does the stone take to reach the bottom of the cliff? How far from the base of the cliff does the stone strike the ground?
2. A steel ball rolls with constant velocity across a tabletop 1.2 meters high. It rolls off and hits the ground 0.6 meters horizontally from the edge of the table. How fast was the ball rolling?
3. A ball is launched with a velocity of 6.67 m/s at an angle of 49° above the horizontal. Find how long it took the ball to land, how high the ball flew, and the range of the ball.
|
Horizontal |
Vertical |
|
|
|
|
|
|
1. An airplane traveling 1001 meters above the ocean at 125 km/hr is to drop a box of supplies to shipwrecked victims below. How many seconds before being directly overhead should the box be dropped? What is the horizontal distance between the plane and the victims when the box is dropped?
2. Divers at Acapulco dive from a cliff that is 61 meters high. If the rocks below the cliff extend outward for 23 meters, what is the minimum horizontal velocity a diver must have to clear the rocks safely?
3. A dart player throws a dart horizontally at a speed of 12.4 m/s. The dart hits the board 32 cm below the height from which it was thrown. How far away is the player from the board?
4. An arrow is shot at a 30° angle. It has a velocity of 49 m/s. Determine how high the arrow goes and the horizontal distance it will travel.
5. A pitched ball is hit by a batter at a 45° angle. It just clears the outfield fence, 98 meters away. Find the velocity of the ball when it left the bat. Assume the fence is the same height as the pitch.
6. Trailing by two points, and with only 2 seconds remaining in the basketball game, a player makes a jump-shot at an angle of 60°, giving the ball a velocity of 10 m/s. The ball is released at the height of the basket, 3.05 meters above the floor. YES! It’s a score. How much time is left in the game when the basket is made? Shots made outside a semicircle of 6.02 meter radius from a spot directly beneath the basket are awarded 3 points, while those inside score 2 points. Did the player tie the game or put the team ahead?
7. A baseball is hit at 30 m/s at an angle of 53°. Immediately an outfielder runs 4 m/s toward the infield and catches the ball at the same height it was hit. What was the original distance between the batter and the outfielder?