An empirical fact about nature is that motion in one direction (for example, the horizontal) does not appear to influence aspects of the motion in a perpendicular direction (the vertical). Imagine a coin dropped from shoulder height. The elapsed time for the coin to hit the ground, the rate at which its vertical position is changing, and its vertical acceleration are the same whether you do this experiment in a stationary bus or one traveling down a smooth, level highway at 65 mph. The horizontal motion of the coin does not affect these aspects of its vertical motion.[1]
[1] Actually, at extremely high speeds the horizontal and vertical motions are not independent. At speeds comparable to the speed of light, the interdependence between horizontal and vertical motion (because of time dilation) becomes noticeable.
Thus, to completely describe the motion of an object moving both horizontally and vertically you can first ignore the horizontal motion, and describe only the vertical component of the motion, and then ignore the vertical motion, and describe the horizontal component. Putting these kinematic components together gives you a complete description of the motion. This experimental fact about nature will make analyzing multi-dimensional motion no more conceptually difficult than analyzing one-dimensional motion.
Given this independence between motions in perpendicular directions, the same kinematic concepts and relationships utilized in one-dimensional motion will be utilized for multi-dimensional motion.
The position of an object is its location relative to a well-defined coordinate system. In multi-dimensional situations, however, you must designate coordinate systems for all perpendicular directions of interest. The zero and positive direction for one dimension is completely independent of the zero and positive direction for another direction. The location at which all coordinate system zeros intersect is referred to as the origin of the coordinate system.
The velocity is the rate at which the position is changing. Thus, we will define the velocity component in the vertical direction, for example, as the rate at which the vertical position is changing. The velocity component in the vertical direction is completely independent of the horizontal position or the rate at which the horizontal position changes.
As long as the coordinate directions are perpendicular, the speed, or magnitude of the object’s velocity, can be determined by:
The direction of the object’s velocity can be determined via right-angle trigonometry.
The acceleration is the rate at which the velocity is changing. Thus, we will define the acceleration component in the vertical direction, for example, as the rate at which the velocity component in the vertical direction is changing. The acceleration component in the vertical direction is completely independent of the velocity component in the horizontal direction or the rate at which the velocity component in the horizontal direction changes.
As long as the coordinate directions are perpendicular, the magnitude of the object’s acceleration can always be determined by:
The direction of the object’s acceleration can be determined via right-angle trigonometry
Doing kinematics in multiple dimensions involves a concerted effort on your part to disregard motion in one direction when considering motion in a perpendicular direction. The ability to mentally break down a complicated motion into its component motions requires considerable practice.
Beginning your analysis by drawing a motion diagram is always the correct first step:
In the shot put, a large mass is thrown at an angle of 22o above horizontal, from a position of 2 m above the ground, a horizontal distance of 25 m.
A motion diagram for this scenario is sketched below.
The verbal representation of the situation has already been translated into a pictorial representation, the motion diagram. A careful reading of the motion diagram allows the construction of the motion graphs.
First, examine the position of the shot put as it moves through the air. Remember, the analysis of the horizontal position must be independent of the analysis of the vertical position.
From the motion diagram, the shot put starts at position zero, and then has positive, increasing positions throughout the remainder of its motion. The horizontal position increases by even amounts in even time intervals.
The shot put starts at position zero, increases its vertical position at a rate that is decreasing, then begins to decrease its vertical position at a rate that is increasing, even as it drops to negative positions.
Typically, both the horizontal and vertical positions are displayed on the same axis.
In the horizontal direction, the rate at which the position changes is constant. Hence, the horizontal component of velocity is constant, and positive. In the vertical direction, the velocity component begins positive, decreases to zero, and then increases in the negative direction.
From the motion diagram, the acceleration of the shot-put can be determined to be directed downward at every point. Thus, the horizontal component of acceleration is zero and the vertical component is negative, and approximately constant due to our model’s approximations.
In the shot put, a large mass is thrown at an angle of 22o above horizontal, from a position of 2 m above the ground, a horizontal distance of 25 m.
Now that you have constructed a motion diagram and motion graphs, you should be able to assign numerical values to several of the kinematic variables. A glance at the situation description should indicate that information is presented about the shot put at two distinct events: when the shot put leaves the putter’s hand and when the shot put strikes the ground. Other information can also be determined about these events by referencing the motion diagram. To organize this information, you should construct a motion table.
In addition to the information explicitly given (the initial and final positions), information is available about both the initial velocity and the acceleration.
In Model 1, you were presented with two kinematic relationships. These relationships are valid in both the horizontal and vertical directions. Thus, you have a total of four relationships with which to analyze the scenario given. In the example above, there are four unknown kinematic variables. You should remember from algebra that four equations are sufficient to calculate four unknowns. Thus, by applying the kinematic relations in both the horizontal and vertical directions, you should be able to determine the initial velocity of the shot-put, the time in the air, and the final horizontal and vertical velocity components.
First, let’s examine the horizontal component of the motion. Note that the positions, velocities, and accelerations in the following equations are all horizontal components.
x-direction
Now let’s examine the vertical component of the motion. All the positions, velocities, and accelerations in the following equations are now vertical components.
y-direction
Plugging t2 = 1.57 s into all of the remaining equations gives:
v1 = 17.2 m/s v2x = 15.9 m/s v2y = -8.94 m/s
Let’s look again at the shot-putter.
In the shot put, a large mass is thrown at an angle of 22 o above horizontal, from a position of 2 m above the ground, a horizontal distance of 25 m.
Imagine a video of the shot put event. Fast-forward over the frames showing the shot putter picking up the shot and stepping into the ring. Begin to watch the imaginary video frame-by-frame as the shot putter begins to push the shot off of her shoulder and forward. Stop the video on the frame when the shot first leaves the putter’s hand.
Why is it so important that we begin the analysis at this frame and explicitly disregard all the motion that has taken place before this frame? The reason is that in every frame preceding this frame, the shot put was in contact with the putter. Thus, the putter was exerting a force on the shot. Since no information is presented concerning this force, we have no way to determine the acceleration during these frames and hence no way to determine any other kinematic variables. Thus, we disregard all motion preceding the instant the shot leaves the putter’s hand because that portion of the motion is simply impossible to analyze with the information provided. Once the shot leaves her hand, the only force acting on the shot is the force of gravity, which greatly simplifies the analysis.
Continue playing the imaginary video forward. Begin playing the tape frame-by-frame as the shot approaches the ground. Stop the video the frame before the shot hits the ground. We will stop our analysis at this frame. Why? Because starting with the next frame, the shot is in contact with the ground. Once in contact with the ground, an additional, unknown magnitude force begins to act on the shot. Once an unknown magnitude force begins to act, the acceleration of the shot becomes unknown and we are stuck. Thus, we conveniently stop our analysis before things get too complicated!
Since our analysis stops the instant before contact, note that the shot is still moving at this instant. (If it wasn’t, how could it ever reach the ground?) Thus, resist the temptation to think that the velocity of the shot is zero at the end of analysis. The velocity of the shot is ultimately equal to zero (after it makes a big divot into the ground) but that happens long after it strikes the ground and hence long after our analysis is finished.
For each of the motion diagrams below, determine the algebraic sign (+, – or zero) of the x- and y-position, velocity, and acceleration of the object at location of the three open circles.
