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9. Parabolas

In the section on Speed, we studied the motion of objects or people traveling at constant speeds. In the section on vectors, we learned how forces acting in different directions create motion in a particular path. By building on these two concepts, we can understand one of the most common phenomena in sports - how objects travel in the air.

From basketballs to high jumpers to shot puts, nearly all objects travel through the air in the same kind of path. The path is called a parabolic curve or a parabola. Parabolas and quadratic equations, the equations that describe their paths, occur in many different places.

9.1 Parabolic Travel

To understand parabolic curves, we need to separate the horizontal and vertical motion. In the horizontal direction, an object travels at a nearly constant speed. Air friction causes it to slow down (which weíll ignore here). In the vertical direction, gravity causes an object to slow down on its way up and then speed up on its way down.

9.1.1 Gravitational Acceleration

Gravity accelerates objects at a relatively constant rate. The rate is known as the gravitational constant. In the US system of measurements, the gravitational constant, g, is 32.2 ft/sec². In metric units, g is 9.81 meters/sec.². Here are two equations involving gravitational acceleration:

(9-1)

(9-2)

where d = the vertical distance

a = acceleration

t = time

v = velocity

Equation 9-1 tells us how far an object will fall in t seconds. Equation 9-2 determines how fast it will be traveling in t seconds. To see how gravity accelerates an object click the figure above. Notice how with every half second, the object travels farther. To understand the equations for gravity, letπs look at the travel of a high diver.

Example 9.1: Brenda performs her dive from a platform 50 feet above the water level. How long will it take her to hit the water and what will her velocity be when she hits the water?

Solution: To determine the time, we use Equation 9-1 and solve for t.

where d = distance

t = time

a = acceleration = g = 32.2 ft/sec.²

We know the time is positive, so we can disregard the negative value.

It takes 1.76 seconds for Brenda to hit the water.

To determine her velocity when she hits the water, we put 1.76 seconds into Eq. 9-2.

= gt = (32.2 ft./sec.²) (1.76 sec.) = 56.7 ft./sec

When Brenda hits the water, she is traveling at 56.7 feet per second.

Note that Brenda's velocity increases with time. The fact that she's traveling faster and faster is why g is known as the acceleration of gravity. The figure below shows the distances an object will fall every half second.

9.1.2 Parabolic Trajectories

When a ball flies into the air, it starts with a vertical and horizontal velocity. Gravity combines with these velocities and causes the ball to travel in a parabolic arc. To see how gravity causes the ball to travel in an arc, click the picture below. The figures show the movement of the ball from three different views, from overhead, from the side, and from the end of the arc.

By combining the acceleration of gravity with the velocity of the object when it is first thrown, formulas can be derived which show the path of the object as it moves through the air. The following formulas give the position and velocities of an object as a function of time:

(9-3) (9-5)

(9-4) (9-6)

where = the initial velocity of the object in the x-direction

= the initial velocity of the object in the y-direction

g = the gravitational constant

t = the time elapsed since the object was thrown

The 0's in the terms mean that the terms represent conditions at time equals zero. The and terms are the velocity components of the object when it first starts moving. In the xy plane, the initial velocity vector could be written as i + j.

We assume that the velocity in the x-direction stays constant. Note that Eq. 9-3 basically identical to the d = rt equation discussed in the Speed chapter.

The velocity in the y-direction, , changes because of the pull of gravity. In the equation , the term" " represents the height the object would be at without gravity. The " " term is the distance gravity has pulled it back. If the travel of an object is plotted on an xy plane, its position is given by the equation:

(9-7)

Other equations include:

whereH = the maximum height reached by the object (9-8)

where R = the range of travel (9-9)

Equations 9-8 and 9-9 assume that the object begins its travel at the ground level and that the land in the area is flat. To determine heights achieved by objects starting their travel above the ground (like a baseball being hit), you need to add the height of the point where the travel begins.

For flexible objects like a human body or for spinning objects, it is the center of gravity of the object that stays on a constant parabolic path.

Example 9.2: During a football game, a field goal kicker kicks the ball with an initial upward velocity (v0y) of 46 ft./sec. and an initial horizontal velocity (v0x) of 46 ft./sec. The ball is kicked from the 30 yard line and is kicked on-line (in the direction between the goal posts). No wind affects the ball.

a.) What is the equation of the vertical path of the object?

b.) Plot the path of the ball.

c.) What is the height of the ball at 1 second?

d.) What is the maximum height (H) reached by the ball?

e.) If the ball is not stopped before it hits the ground, what is the horizontal distance (R) that the ball travels?

f) Is the field goal good?

Solution: If the ball is kicked from the 30-yard line, it is 40 yards from the goal posts. (The end zone is 10 yards deep.) Since the gravitational constant is given in units of ft./sec.², we need to convert the distances from yards to feet. A distance of 40 yards is equivalent to 120 feet.

a.) We use Equation 9-7 to find the path of the object,

with x and y in ft.