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Conceptual Problem about Projectile Motion Learning Goal: To understand projectile motion by considering horizontal constant velocity motion and vertical constant acceleration motion independently. Projectile motion refers to the motion of unpowered objects (called projectiles) such as balls or stones moving near the surface of the earth under the influence of the earth's gravity alone. In this analysis we assume that air resistance can be neglected. An object undergoing projectile motion near the surface of the earth obeys the following rules: An object undergoing projectile motion travels horizontally at a constant rate. That is, the x component of its velocity, , is constant. An object undergoing projectile motion moves vertically with a constant downward acceleration whose magnitude, denoted by , is equal to 9.80 near the surface of the earth. Hence, the y component of its velocity, , changes continuously. An object undergoing projectile motion will undergo the horizontal and vertical motions described above from the instant it is launched until the instant it strikes the ground again. Even though the horizontal and vertical motions can be treated independently, they are related by the fact that they occur for exactly the same amount of time, namely the time the projectile is in the air. The figure shows the trajectory (i.e., the path) of a ball undergoing projectile motion over level ground. The time corresponds to the moment just after the ball is launched from position and . Its launch velocity, also called the initial velocity, is . Two other points along the trajectory are indicated in the figure. One is the moment the ball reaches the peak of its trajectory, at time with velocity . Its position at this moment is denoted by or since it is at its maximum height. The other point, at time with velocity , corresponds to the moment just before the ball strikes the ground on the way back down. At this time its position is , also known as ( since it is at its maximum horizontal range. Projectile motion is symmetric about the peak, provided the object lands at the same vertical height from which is was launched, as is the case here. Hence . Consider a diagram of the ball at time . Recall that refers to the instant just after the ball has been launched, so it is still at ground level (). However, it is already moving with initial velocity , whose magnitude is and direction is counterclockwise from the positive x direction. The peak of the trajectory occurs at time . This is the point where the ball reaches its maximum height . At the peak the ball switches from moving up to moving down, even as it continues to travel horizontally at a constant rate. The flight time refers to the total amount of time the ball is in the air, from just after it is launched () until just before it lands (). Hence the flight time can be calculated as , or just in this particular situation since . Because the ball lands at the same height from which it was launched, by symmetry it spends half its flight time traveling up to the peak and the other half traveling back down. The flight time is determined by the initial vertical component of the velocity and by the acceleration. The flight time does not depend on whether the object is moving horizontally while it is in the air. The range of the ball refers to how far it moves horizontally, from just after it is launched until just before it lands. Range is defined as , or just in this particular situation since . Range can be calculated as the product of the flight time and the x component of the velocity (which is the same at all times, so ). The value of can be found from the launch speed and the launch angle using trigonometric functions, as was done in Part B. The flight time is related to the initial y component of the velocity, which may also be found from and using trig functions. The following equations may be useful in solving projectile motion problems, but these equations apply only to a projectile launched over level ground from position () at time with initial speed and launch angle measured from the horizontal. As was the case above, refers to the flight time and refers to the range of the projectile. flight time: range: In general, a high launch angle yields a long flight time but a small horizontal speed and hence little range. A low launch angle gives a larger horizontal speed, but less flight time in which to accumulate range. The launch angle that achieves the maximum range for projectile motion over level ground is 45 degrees. Projectile Motion Tutorial Learning Goal: Understand how to apply the equations for 1-dimensional motion to the y and x directions separately in order to derive standard formulae for the range and height of a projectile. A projectile is fired from ground level at time , at an angle with respect to the horizontal. It has an initial speed . In this problem we are assuming that the ground is level. Arrow Hits Apple An arrow is shot at an angle of above the horizontal. The arrow hits a tree a horizontal distance away, at the same height above the ground as it was shot. Use for the magnitude of the acceleration due to gravity. Suppose someone drops an apple from a vertical distance of 6.0 meters, directly above the point where the arrow hits the tree.

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