Question

In ideal projectile motion, the velocity and acceleration of the projectile at its maximum height are, respectively, a) horizontal, vertical downward. b) horizontal, zero. c) zero, zero. d) zero, vertical downward. e) zero, horizontal.

Short answer

Answer: a) horizontal, vertical downward

Step-by-step solution

Understanding Ideal Projectile Motion

Ideal projectile motion is a two-dimensional motion under the influence of gravity. It is called "ideal" because it assumes no air resistance or other external forces except gravity. In this motion, an object is launched at an angle to the horizontal, and its path traces a parabolic trajectory.

Finding Velocity at Maximum Height

At the maximum height, the vertical component of the projectile's velocity becomes zero because the object has reached its highest point and is not moving upward or downward momentarily. However, the horizontal component of the velocity remains constant throughout the motion, as there are no external forces acting on it in the horizontal direction. Hence, at the maximum height, the velocity of the projectile is horizontal.

Finding Acceleration at Maximum Height

Acceleration in ideal projectile motion is constant and is always directed downward, due to gravity. This acceleration is known as the gravitational acceleration (g) and is usually approximated to 9.81 m/s². It is essential to note that acceleration due to gravity always acts on the projectile, even at its maximum height. Thus, the acceleration of the projectile at its maximum height is vertical downward. Using the information provided in steps 2 and 3, we can now answer the exercise question. The velocity of the projectile at its maximum height is horizontal, and its acceleration is vertical downward. Therefore, the correct answer is: a) horizontal, vertical downward.

Key concepts

Velocity at Maximum Height

When discussing ideal projectile motion, a crucial point to understand is the projectile's velocity at its maximum height. Imagine throwing a ball at an upward angle; as it rises, it slows down due to gravity. At the peak of its ascent, known as the maximum height, the vertical component of its velocity is momentarily nil. It's at this point that the ball is neither climbing nor falling; it's paused in the vertical sense.
However, the horizontal component of its velocity doesn't cease. It persists unchanged because, in an ideal scenario, we disregard air resistance and other horizontal forces. Thus, at the zenith of its trajectory, the projectile sails onward solely with a horizontal velocity. This simplistic view of motion helps break down the complex journey of a projectile into manageable concepts, aiding students in their understanding of an object in flight.

Acceleration Due to Gravity

The acceleration due to gravity is an ever-present influence in projectile motion. Regardless of the projectile's position—whether ascending, descending, or at its apex—gravity exerts a consistent pull downward. This force is reflected as an acceleration, directly toward the Earth's center, and is symbolized as 'g', averaging approximately 9.81 m/s².
Students might wonder, 'Does gravity cease at the top?' The answer is a definitive 'no'. Even when the projectile reaches its maximum height and the upward movement halts, gravity is still at work, ensuring that the projectile will resume its descent. Hence, throughout the entire trajectory, the projectile retains this unwavering downward acceleration, underlining gravity's relentless influence on terrestrial objects.

Parabolic Trajectory

The term parabolic trajectory is often associated with ideal projectile motion, illustrating the graceful arc a projectile describes when launched at an angle into a gravity field with no air resistance. Why is it parabolic? It's derived from the fact that the motion of the projectile, influenced by a consistent gravitational pull, forms a shape that is symmetrically curved—a shape known as a parabola.
Visualize a basketball player making a jump shot; the ball arcs through the air, climbing, peaking, then descending into the hoop, assuming a parabolic path. This is not just an aesthetic marvel; it’s a fundamental concept demonstrating kinetic energy conversion into potential energy and back, all under the influence of gravity. Understanding the parabolic trajectory enhances a student's grasp of the forces at play and how energy is conserved and transformed in an isolated system like ideal projectile motion.