4.0 - Introduction
Imagine that you are standing on the 86th floor observatory of the Empire State
Building, holding a baseball. A friend waits in the street below, ready to catch the
ball. You toss it forward and watch it move in that direction at the same time as it
plummets toward the ground. Although you have not thrown the ball downward at
all, common sense tells you that your friend had better be wearing a well-padded
glove!
When you tossed the ball, you subconsciously split its movement into two
dimensions. You supplied the initial forward velocity that caused the ball to move
out toward the street. You did not have to supply any downward vertical velocity.
The force of gravity did that for you, accelerating the ball toward the ground. If you
had wanted to, you could have simply leaned over and dropped the ball off the roof,
supplying no initial velocity at all and allowing gravity to take over.
To understand the baseball’s motion, you need to analyze it in two dimensions.
Physicists use x and y coordinates to discuss the horizontal and vertical motion of
the ball. In the horizontal direction, along the x axis, you supply the initial forward
velocity to the ball. In the vertical direction, along the y axis, gravity does the work.
The ball’s vertical velocity is completely independent of its horizontal velocity. In
fact, the ball will land on the ground at the same time regardless of whether you
drop it straight off the building or hurl it forward at a Randy Johnson-esque 98 miles
per hour.
To get a feel for motion in two dimensions, run the simulations on the right. In the
first simulation, you try to drive a race car around a circular track by controlling its x
and y component velocities separately, using the arrow keys on your keyboard. The
right arrow increases the x velocity and the left arrow decreases it. The up arrow
key increases the y velocity, and yes, the down arrow decreases it. Your mission is
to stay on the course and, if possible, complete a lap using these keys.
Your car will start moving when you press any of the arrow keys. On the gauges,
you can observe the x and y velocities of your car, as well as its overall speed.
Does changing the x velocity affect the y velocity, or vice-versa? How do the two
velocities seem to relate to the overall speed?
There is also a clock, so you can see which among your friends gets the car around
the track in the shortest time. There is no penalty for driving your car off the track,
though striking a wall is not good for your insurance rates. Press RESET to start over. Happy motoring!
In the second simulation, you can experiment with motion in two dimensions by firing the cannon from the castle. The cannon fires the
cannonball horizontally from the top of a tower. You change the horizontal velocity of the cannonball by dragging the head of the arrow. Try to
hit the two haystacks on the plain to see who is hiding inside.
As the cannonball moves, look at the gauges in the control panel. One displays the horizontal velocity of the cannonball, its displacement per
unit time along the x axis. The other gauge displays the cannonball’s vertical velocity, its displacement per unit time along the y axis. As you
use the simulation, consider these important questions: Does the cannonball’s horizontal velocity change as it moves through the air? Does its
vertical velocity change? The simulation pauses when a cannonball hits the ground, and the gauges display the values from an instant before
that moment.
The simulation also contains a timer that starts when the ball is fired and stops when it hits a haystack or the ground. Note the values in the
timer as you fire shots of varying horizontal velocity. Does the ball stay in the air longer if you increase the horizontal velocity, or does it stay in
the air the same amount of time regardless of that velocity?
The answers to these questions are the keys to understanding what is called projectile motion, motion where the acceleration occurs due to
gravity alone. This chapter will introduce you to motion in two and three dimensions; projectile motion is one example of this type of motion.