Texas Lutheran University Department of Physics Seguin, TX 78155

Texas Lutheran UniversityDepartment of Physics

Seguin, TX 781551

Bulldog PhysicsBulldog Physics

Two-Dimensional KinematicsTwo-Dimensional Kinematics

0fv v at

2 20 02 ( )fv v a x x

20 0

1

2x x v t at


Seguin, TX 781552

Projectile MotionWhat is a Projectile? x

y

Projectile motion is a example of 2D MotionAny object that flies through the air.

An object flying through the air moves in both the x and y directions.

Typically, the motion in each component is uncoupled.The motion in each component is independent of the other.

What is the acceleration of an object while it’s flying through the air?

Acceleration due to gravity! ˆa g j

What is the acceleration in the x-direction?

It is ZERO!!! 0xa What does this mean about the velocity in the x-direction?

It is constant.

A blue ball is being held above ground a distance y. A red ball is thrown from the same height, y, at a horizontal velocity of v0. The blue ball is released at the same instant the red ball leaves the table. Which ball reaches the ground first?

1) The blue ball lands first.

2) The red ball lands first.

3) They land simultaneously.

4) There is not enough information.


Seguin, TX 781553

Conceptual Question 1

0v

y

A blue ball is being held above ground a distance y. A red ball is thrown from the same height, y, at a horizontal velocity of v0. The blue ball is released at the same instant the red ball leaves the table. Which ball reaches the ground first?

1) The blue ball lands first.

2) The red ball lands first.

3) They land simultaneously.



Seguin, TX 781554


0v

y


Seguin, TX 781555

A Ball in the AirWe can separate the motion in the x and y directions.

x

y

x-direction y-direction

xa 0 ya g

0,xv 0 cosv 0, yv 0 sinv

0v

x L y 0

L

We also can talk about the time in the air.This must be the same for both directions!So t can go in both columns.

t t

Now we use the kinematics equations using only one column at a time to solve our problems.

What is the velocity at max height?

xa 0, so vx is constant.

So only vy is zero at the maximum height.


Seguin, TX 781556

Example Problem 1 Long TD Pass video


Seguin, TX 781557

Example Problem 1

A quarterback throws a football at 45 mph (20.1 m/s) at 35 degrees with the horizontal to get the ball over the linemen. Assume the ball is caught at the same height it is thrown.

(a)How long is the ball in the air? (b)How far away can the receiver be and catch the ball?(c)What is the maximum height of the ball while in the air?

Play “Long TD Pass”


Seguin, TX 781558

A Ball in the Air (Horizontal Velocity Case)We still have the same basic columns. x

y


xa 0 ya g

0,xv 0v 0, yv 0

0v

x L y h

L

We now have a change in height, so y is no longer zero.t t

Again, we’re now set up to solve problems using the kinematics equations using one column at a time.

h

Our initial velocity is now completely in the x-direction.


Seguin, TX 781559

Example Problem 2

A thirdbaseman catches a ground ball and throws to the firstbaseman, who is 38.0 meters away. The thirdbaseman throws the ball horiztonally at height of 1.83 meters off the ground. What is the minimum speed the thirdbaseman can throw the ball without the ball hitting the ground before reaching the firstbaseman?


Seguin, TX 7815510

A Ball in the Air (Angle and Height)We still have the same basic columns. x

y


xa 0 ya g

0,xv 0 cosv 0, yv 0 sinv x L y h

L

When this situation arises, we almost always get a quadratic equation when analyzing the y-direction.

t t

h

0v

20 0,

1

2y yy y v t a t

20 sin

2

gh v t t

20 sin 0

2

gt v t h

How do we solve this?


Seguin, TX 7815511

The Quadratic EquationWe use an algebra trick:

20 sin 0

2

gt v t h

If we have an equation of the form: 2 0At Bt C

Then the solution is:2 4

2

B B ACt

A

For the situation at hand, we have 2

gA

0 sinB v

C hSo we get

2 20 0sin sin 4

2

22

gv v h

tg

2 20 0sin sin 2v v gh

tg

The plus and minus give you up to two possible solutions. This means there are two times where you reach the specified height.


Seguin, TX 7815512

Example Problem 3

A golfer tees off on a par 3 course. The she lofts the ball at a 44 degree angle from the horizontal, sending it off at 24 m/s. The hole is positioned 50 feet (15.2 meters) below the tee. The ball lands right into the hole! (a)What is the horizontal distance between the tee and the hole? (b)How fast the ball traveling just before it lands in the hole?

A quarterback throws two passes, one to Receiver A and the second to Receiver B. The trajectories are shown. Which pass takes more time to get to the receiver?

1) The pass to Receiver A.

2) The pass to Receiver B.

3) Both passes spend the same time in the air.



Seguin, TX 7815513


A quarterback throws two passes, one to Receiver A and the second to Receiver B. The trajectories are shown. Which pass takes more time to get to the receiver?

1) The pass to Receiver A.

2) The pass to Receiver B.

3) Both passes spend the same time in the air.



Seguin, TX 7815514


Receiver A Receiver B

Texas Lutheran University Department of Physics Seguin, TX 78155

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