Break problems down into 1-d components

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1 Motion in 2-d Up until now, we have only been dealing with motion in one-dimension. However, now we have the tools in place to deal with motion in multiple dimensions. We have seen how vectors can be broken into components, and we have made general vector definitions of displacement, velocity, and acceleration. The trick to dealing with more than one dimension will always be the same Break problems down into 1-d components Although somewhat surprising, we can deal with only the components of motion. For instance, although an object may be moving both in the horizontal and vertical directions simultaneously, it turns out that we can analyze each of these directions independently. This approach is incredibly powerful, and allows us to solve many seemingly complicated problems. A Comment on Air Resistance Air resistance is a completely real and important effect, whether designing a car, or positioning yourself most efficiently on a bicycle. However, in many cases the effect of air resistance is small and can be ignored. If you ever hold your hand out of a car window, you feel very little force on your hand at low speeds, and the effect is only obvious at higher speeds. In fact, the effect depends on the way you hold your hand; it feels different if you make a fist or if you put your palm flat or sideways. Obviously, air resistance is a complicated effect that depends on both the speed of an object and its shape, so it is beyond the scope of our current studies. Thus, we choose situations in which this effect is small, or would not change the essence of a problem. Projectile Motion One of the classic applications of solving kinematic problems in 2-d is projectile motion. By projectile motion, we mean some object flying through the air that has been given an initial velocity but while in the air it is only subject to the acceleration due to gravity, and no other accelerations. Again, we will be ignoring air resistance for the following explanations and examples. Below is a schematic of a general projectile motion situation

2 We should take note of several important features of this problem: When we are given some parameters of the initial velocity (or the velocity at another time in the problem), we are being shown a snapshot of the motion. For instance, in the diagram shown we are given the velocity of the projectile when it leaves the ground, and the angle relative to the ground. Again, the vector shown is just a snapshot at a particular time, the direction of velocity changes continuously (and is tangent to the trajectory of the projectile). In general, the projectile does not need to take off and land at the same vertical position. For example, it could be launched from a cliff, or land in hole. We can solve for many different quantities in these types of problems. For example, depending on the information given, we can solve for the projectile s maximum height, its time in the air, the angle from which it was launched, the angle at which it lands, the velocity at any point in the motion, the components of the velocity at any point in the motion, the horizontal distance it travels, or its horizontal and vertical position at any time during the motion. Acceleration is a vector, and can have any direction. But in the special case of acceleration due solely to gravity, the acceleration is always straight down. (In a sense, the idea of down really comes from the direction objects fall). For instance, in the diagram below up has been chosen as the positive direction, and the y-component of the acceleration is given by g. y a ax = 0 ay = g x The equations of motion for projectile motion are exactly like the 1-d motion case, except now we have separate equations for x-motion and y-motion. We can treat the x-motion and y-motion separately. Notice that the equations become particularly simple in the case of the x-direction. Since there is no acceleration in the x-direction, the x-component of the velocity is equal to the initial velocity in the x- direction for the entire motion.

3 Example: Horizontal Rifle A rifle bullet is fired horizontally with an initial velocity of 100 m/s from an initial height of 2.0 m. How much time passes before the bullet hits the ground? What horizontal distance does the bullet travel before it hits the ground? The key idea in all projectile motion problems is that we can treat the x-motions and y-motions separately! The motion along the y-direction (vertical) is independent of the motion along the x- direction (horizontal). For this problem I will choose to the right as the positive x-direction, and since both the acceleration and displacement in the vertical are going down, I will choose down as the positive y-direction. Remember, in 1-d I can treat the plus or minus sign as the directional information of a vector. The time to hit the ground is entirely controlled by the y-motion. We apply our equations of motion first purely in the y-direction. Now we fill in for any values we know are zero, the variables specific to the problem and the directional information (plus or minus signs). Rearranging, we can solve for the time. Plugging in the values we find that ( ) ( )

4 Now we look at the x-direction to see how far along the horizontal direction the bullet traveled in this time (0.639 s). Plugging in our values we find Another interesting question we could ask is What is the speed of the bullet as it falls? Notice, that when we say the speed, we mean the magnitude of the velocity. The magnitude of the velocity is not the magnitude of just the x-component or the y-component; it is the magnitude of the total velocity. As the bullet travels, the x-component of its velocity remains constant, while the y-component (which began as zero in this case) grows larger and larger as the bullet falls. [ ] [ ] Since is the moment when the bullet leaves the gun, and at this instant the initial velocity in the y- direction is zero, the speed is a minimum at. With each passing moment the bullet gains velocity in the y-direction as it falls. Therefore, in this case, the speed is at its maximum when is maximum, which is the moment just before the bullet hits the ground.

5 Example: Minimum Speed of a Projectile Fired at an Angle A projectile is fired at with initial speed at an angle above the horizontal. What is the minimum speed of the projectile? We know that since we are considering cases in which the air resistance can be ignored, there is no acceleration in the x-direction. Therefore, the x-component of the velocity is constant throughout the motion, and equal to its initial value. However, the y-component of the velocity reaches a minimum at the apex of the motion. At the apex, the y-component of the velocity is zero. So the minimum speed occurs at the apex.