Projectile Motion
An activity from the Burroughs Wellcome Student Research Program The Science of Sports

Problem:

If you throw a baseball through the air you will notice that its path resembles a parabola. This is because gravity is pulling the baseball down but very little is affecting the balls forward motion. With nothing more than a meterstick, can you predict the range of a projectile (such as a ball or dart) at a known angle?

Introduction:

Gravity is a force that acts only on objects with mass. As you know from Newton's 2nd Law, forces cause objects to accelerate or change their velocity with time. Acceleration can be expressed by the formula:

a =	(D v)
	(D t)

(2)

where a is acceleration, D v is the change in velocity and D t is the time interval. By combining this equation with the formula for average velocity:

v =	(D s)
	(D t)

(1)

we get an equation for accelerated motion:

D s = v_o t + ¹/₂ a t²
(3)

where s is displacement and vo is the initial velocity. You might recognize this from the Reaction Time Acitvity where vo was zero because the meter stick was dropped from rest and the acceleration was g = 9.8 m/s/s - the acceleration due to gravity.

Also if you look carefully at eqation (3) you can see that if there is no acceleration, it becomes exactly the same as equation (2)

All three of these equations work if the motion is in one dimension such as dropping a meter stick or throwing a ball straight up. However, if you wish to look at motion in two dimensions such as the flight of an arrow or a fly ball you must remember one simple fact. Gravity ONLY works vertically or in the y direction. Gravity does not pull things sideways, it only pulls them down. Therefore when motion is horizontal (x direction) and vertical (y direction) at the same time we must consider them seperately because gravity ONLY affects the y part of the motion.

Separating the x and y parts of an objects velocity is not difficult. Consider a simple right triangle.

Using simple trigonometry we know that sinq= v_y/v or v_y= v sinq and cosq= v_x/v or v_x = vcosq. When an object is moving at an angle q, we can consider the x and y dimensions seperately and still use the equations above (1-3).

An easy example:
Suppose a football is kicked at at 30^{o angle with a velocity of 15 m/s and we need to know how high it goes to see if it will clear the field goal.}

What do you know?	At the beginning vy = 15 m/s sin 30o = 7.5 m/s and vx = 15 m/s cos 30o = 13 m/s At the very top vy = 0 m/s because it stops moving upward. Using this we know that Dvy = -7.5 m/s (the velocity decreases by 7.5 m/s) For the whole trip a = -9.8 m/s/s
What are you trying to find?	The height of the ball or the y displacement
Which equations have this stuff in them?	equation 1: a = (D v) (D t) equation 3: D s = vo t + 1/2 a t2 Both equations also have time in them which we do not know but we can use equation 1 to solve for time and then plug that in equation 3.
Let's solve the problem!	-9.8 m/s/s = -7.5 m/s Dt solving for Dt give Dt =.76 seconds This is the time it takes the ball to reach the highest point. Now we can find the height using equation 3. Ds = 7.5 m/s (.76s) + 1/2 (-9.8 m/s/s)(.76s)2 Solving for Ds gives Ds =2.9 m Does the ball make the field goal?

Materials
Dart Gun (Nerf makes a good one) protractor meterstick

^{Exploring I}

Find the total velocity of a dart upon exiting your gun. Some hints: If you only have a meterstick, what things can you measure?
Which directions should you shoot your gun so that the x or y velocity is zero?
Of the two directions you can shoot the gun, which one gives values easy to measure with a meter stick?

Discussion I:

What is the total intial velocity of your dart gun? (Show all your work)

Exploring II

Give your dart to the teacher. Using the information collected above and the equations, predict how far way the dart will land if shot from the angle specified on your gun. Some hints: Is your dart traveling in one dimension only or two at the same time?
In what dimension is the value you are trying to find?
Make sure that you list everything you know.
What is the only difference between take off and landing?

Discussion II:

• How far away do you predict your dart will land? Show all your work.
  Check this with a teacher before attempting to shoot the dart gun.

• How far away does the dart actually land? (best of 2 tries)

• What angle do you think you should shoot you shoot a dart gun (or hit a baseball) to obtain the maximum range or x displacement? Can you prove it?

• Which ball do you think will go further -
  1) a ball hit at 10 m/s and an angle of 30o
  or 2) a ball hit at 10 m/s and 60o? Can you prove it?

• What factors in the real world (with air) might affect the height, distance or time of flight of a baseball?