Two Dimensional Kinematics
Objectives:
1. Sketch the theoretical path of a projectile.
2. Solve problems involving projectile motion for projectiles fired
horizontally and at angles.
3. Solve problems involving changing frames of reference and
relative velocities.
4
Projectile motion can be understood by analyzing the x (horizontal) and y (vertical) components of a
vector.
𝐴𝑦 = 𝐴𝑠𝑖𝑛𝜃
𝐴𝑥 = 𝐴𝑐𝑜𝑠𝜃
Galileo predicted that an object projected horizontally will reach the ground _____faster than/slower
than/at the same time as________ an object dropped vertically from the same height.
When an object is projected at an angle, the vertical component (Vy) is _______________ when the
object reaches its highest point.
When an object is projected at an angle, the horizontal component (Vx) remains __________________
throughout the flight because the horizontal acceleration is _____________ m/s2.
Example: A ball rolls off the horizontal roof of a building that is 20 meters from the ground. The ball
lands 15 meters from the bottom of the building. How fast was the ball rolling when it left the building?
A soccer ball is kicked with a velocity of 15 m/s at an angle of 30˚ above the ground.
a.
b.
c.
d.
e.
f.
What is the horizontal component of the velocity?
What is the vertical component of the velocity?
What is the maximum height above the ground that the ball reaches?
How much time is the ball in the air?
How far away does the ball hit the ground?
What is the magnitude and direction of the acceleration of the ball throughout its flight?
Two Dimensional Motion
The results of a vector that acts in the “x” plane are unaffected by the results of a vector in the “y”
plane.
However, the motion that occurs is linked by time.
Example: The River Crossing Problem
A boat traveling due east at 5 m/s crosses a river that flows from north to south at a rate of 3
m/s. The river is 100 m across.
a. How long does it take the boat to cross the river?
b. What is the magnitude of the boat’s displacement at the end of the trip?
c. What is the boats average velocity for the trip?
1. Create a motion map for an object that is dropped from rest.
2. Create a motion map for an object that is moving horizontally with an acceleration of 0.0 m/s2.
3. What is the x-acceleration of an object thrown horizontally in the x-direction? _________
What is the y-acceleration of an object thrown horizontally in the x-direction? _________
Create a motion map for an object that is thrown horizontally in the x-direction. Show x and y
velocity vectors.
Horizontal Projectiles
Velocity vectors
Vy=0
Vx
Vx= same throughout flight b/c ax = 0 m/s2.
Vy increases at a rate of 10 m/s2 because of acceleration due to gravity.
Acceleration Vectors
ay= -10 m/s2 everywhere!
ax = 0 m/s2 **There is no acceleration in the horizontal plane because there is no net force in the
horizontal plane.
Mathematical Analysis of Horizontal Projectiles
*Treat like “river crossing.”
*Velocity in the “x” plane is constant.
*Motion in the “y” plane is accelerated motion. Accelerated by gravity at -9.81 m/s2.
* Motion in the “x” plane occurs at the same time as motion in the “y” plane. The motions are
linked by time.
Example:
A car is traveling at 35 m/s drives off a 50 m cliff.
a. How far from the bottom of the cliff does the car land?
b. With what speed does the car hit the ground?
Horizontal Projectiles
1. A ball is rolled off the roof of a building with a velocity of 15 m/s. The height of the building is
30m.
a. How much time will it take for the ball to hit the ground?
b. How far from the bottom of the building will the ball land?
2. An airplane flying horizontally at 100 m/s drops a bomb from a height of 1000m. How far from
the target must the plane release the bomb?
3. An object is kicked from the roof of a building that is 20 m tall and lands 10 m from the bottom
of the building. With what speed was the object kicked off the building?
4. A physics student sees a ball thrown from the top of a building. The student notices that it takes
the ball 2.5 seconds to hit the ground and that it lands 17 m from the bottom of the building.
a. How high is the building?
b. With what horizontal speed was the ball thrown?
5. A ball is thrown from the roof of a 50 m tall building into the window of an adjacent building
which is 30 m away. The bottom of a window in the second building is 40 m from the ground.
The window is 2.0 m from bottom to top. What are the minimum and maximum horizontal
velocities that the ball must be thrown in order to make it into the window?
Angled Projectile Vector Diagrams
Velocity Vectors:
Important Points:
*Vx is constant throughout flight.
*Vy decreases at a rate of 10 m/s2 on the way up.
*Vy increases at a rate of 10 m/s2 on the way down.
*Vy is zero at the highest point of flight.
Acceleration Vectors:
Important points:
*acceleration is constant throughout
-ay is always -10 m/s2
-ax is always 0 m/s2
Mathematical Analysis of Angled Projectiles
When Projectiles are fired at angles:
-The initial velocity vector must be broken into its horizontal and vertical components.
-The horizontal component remains constant because the acceleration of the plane is 0 m/s2.
-The vertical component is accelerated by gravity at a rate of -10 m/s2.
-Treat the problem like two separate problems: One in the x-plane, and one in the y-plane.
Example 1:
An arrow is fired at an initial velocity of 50 m/s at an angle of 40˚ to the horizontal. Determine:
a.
b.
c.
d.
The horizontal and vertical components of the initial velocity.
The total time the arrow is in the air.
The horizontal distance (range) the arrow travels.
The maximum height of the arrow.
Example 2:
A golf ball is hit to a green that is a horizontal distance of 100 m away and is elevated 10 m abve
the golfer’s feet. The ball lands on the green 3 seconds after it is struck with the club.
Determine:
a. The launch speed and angle for the ball.
b. The impact speed of the ball when it hits the green.
Angled Projectile Problems
1. A projectile is fired from a cannon at an angle of 30˚ above the horizontal at 100 m/s. Find the:
a. Horizontal component of the velocity.
b. Vertical component of the velocity.
c. Time that the projectile is in the air (assuming level ground).
d. Horizontal distance covered by the projectile.
e. Maximum height of the projectile.
f. Acceleration of the projectile 2 seconds into its flight.
2. A football is punted and has a hang time of 4.2 seconds. If the punt is 50 meters in the
horizontal direction, what was the initial velocity of the punt?
3. Tiger Powlin uses his lob wedge to hit a spectacular golf shot. The golf ball is given a velocity of
30 m/s at an angle of 65˚ with the ground. If the ball lands on a green that is elevated 13 meters
above Tiger’s feet:
a. How long is the ball in the air?
b. How far away from the green (horizontally) is Tiger when he makes his shot?
4. A projectile is fired with an initial speed of 113 m/s at an angle of 60˚ above the horizontal from
the top of a cliff that is 49.0 meters high. Determine
a. Time to reach maximum height.
b. Maximum height above the base of the cliff.
c. Total time in the air.
d. Horizontal distance from the base of the cliff.
5. A ball is thrown from the top of a building that is 30 m tall. The ball is thrown with a velocity of
10 m/s at an angle of 20˚ below the horizontal. Determine the distance from the bottom of the
building that the ball lands.
6. A ball is thrown at an angle so that its horizontal velocity is 10 m/s and its vertical velocity is 12
m/s. It takes 1.22 seconds for the ball to reach its highest point.
a. What was the launch speed of the ball?
b. What horizontal distance does the ball travel during its entire time in the air?
7. EJ Manual throws a football at an angle of 20˚ to the horizontal. The ball lands 30 meters down
field 1.5 seconds after it was thrown.
a. What is the horizontal component of its initial velocity?
b. What was the initial launch speed of the football?
8. A golf ball leaves a golf club with an initial velocity of 30 m/s at an angle of 50˚ with the
horizontal.
a. What is the vertical component of the golf ball’s initial velocity?
b. What is the total flight time of the ball?
c. What is the total horizontal distance traveled by the golf ball?
9. A child kicks a kickball with an initial velocity of 10 m/s at an angle of 35˚ with the horizontal.
a. What are the horizontal and vertical components of the initial velocity?
b. What is the total time the kickball is in the air?
c. What is the maximum height of the ball?
a. Determine the time it takes for the ball to reach the plane of the fence.
b. Will the ball hit the fence? If so, how far below the top of the fence will it hit? If not, how far
above the fence will it pass?
c. On the axes below, sketch the horizontal and vertical components of the velocity of the ball as
functions of time until the ball reaches the plane of the fence.
Relative Velocity
What is the velocity of the person on the train with respect to the observer on the ground?
What is the velocity of car A with respect to car B?
What is the velocity of car B with respect to car A?
Example 3:
Car A is traveling at 15 m/s due east. Car B is traveling 20 m/s due north. What is the velocity of
car A with respect to car B?
Challenge Question: The rear window of a car is at an angle of 40˚ with the vertical plane. The
car is traveling at 30 m/s. Rain is falling straight down at 20 m/s. Will the rain hit the rear
window? Prove your answer mathematically.