Projectile motion is a fundamental concept in physics that describes the motion of objects launched into the air. When an object is thrown or launched at an angle, it follows a curved path called a parabola. This motion occurs under the influence of gravity alone, assuming we ignore air resistance. Common examples include basketball shots, cannon balls, and water from fountains. The key insight is that projectile motion can be analyzed as two separate components: horizontal motion with constant velocity, and vertical motion with constant acceleration due to gravity.
To understand why projectiles follow curved paths, we must analyze the forces acting on them. When we ignore air resistance, only one force acts on a projectile: gravity. This gravitational force acts vertically downward with magnitude mg, where m is the mass and g is gravitational acceleration. Importantly, no horizontal forces act on the projectile once it's launched. This force analysis reveals why projectile motion has two distinct components. The horizontal motion maintains constant velocity because no horizontal forces act to change it. The vertical motion is accelerated motion because gravity continuously acts downward, changing the vertical velocity over time.
Now let's develop the mathematical framework for projectile motion. We start with basic kinematic equations and apply them to both horizontal and vertical components. For horizontal motion, since no forces act horizontally, we have constant velocity. The position equation is x equals v-zero-x times t, and velocity remains constant at v-zero-x. For vertical motion, gravity causes constant acceleration downward. The position equation becomes y equals v-zero-y times t minus one-half g t squared, and velocity changes according to v-y equals v-zero-y minus g t. The initial velocity v-zero is decomposed into components: v-zero-x equals v-zero cosine theta, and v-zero-y equals v-zero sine theta, where theta is the launch angle.
Now let's derive the trajectory equation and analyze the parabolic path. By eliminating time from our kinematic equations, we get the trajectory equation: y equals x tangent theta minus g x squared over 2 v-zero squared cosine squared theta. This equation describes the parabolic path of any projectile. Key points include: maximum height occurs at time t equals v-zero sine theta over g, and the range is R equals v-zero squared sine 2 theta over g. Different launch angles create different trajectory shapes. Notice how 30 degrees and 60 degrees produce the same range, while 45 degrees gives the maximum range for a given initial velocity.
Now let's explore the relationship between launch angle and range to find the optimal angle. The range equation R equals v-zero squared sine 2 theta over g shows that range depends on sine of twice the angle. Maximum range occurs when sine 2 theta equals 1, which happens when 2 theta equals 90 degrees, giving us the optimal angle of 45 degrees. An interesting property is that complementary angles produce the same range. For example, 30 degrees and 60 degrees give identical ranges, as do 15 degrees and 75 degrees. This happens because sine of 60 degrees equals sine of 120 degrees. The graph clearly shows that 45 degrees maximizes the range for any given initial velocity.