Question

In ideal projectile motion, when the positive $y$ -axis is chosen to be vertically upward, the $y$ -component of the acceleration of the object during the ascending part of the motion and the $y$ -component of the acceleration during the descending part of the motion are, respectively, a) positive, negative. c) positive, positive. b) negative, positive. d) negative, negative.

Short answer

Answer: In ideal projectile motion, the y-component of the acceleration of an object during both the ascending and descending parts of the motion is negative due to the constant influence of gravity acting downward.

Step-by-step solution

Projectile Motion

In ideal projectile motion, an object follows a parabolic trajectory under the influence of gravity alone, with no other forces acting on it, such as air resistance. As the object moves, its velocity vector can be decomposed into two perpendicular components: the horizontal velocity ($v_x$) and the vertical velocity ($v_y$). The horizontal component of velocity remains constant due to the absence of external forces, while the vertical component changes due to the acceleration due to gravity acting in the vertical direction.

Acceleration Due to Gravity

The only force acting on the projectile in this case is gravity, which acts downward. The acceleration due to gravity is constant, with a magnitude of $9.81m{s}^{-2}$ approximately, and it acts downward parallel to the vertical ($y$) axis. Therefore, the $y$-component of the acceleration of the object during both the ascending and descending parts of the motion are the same - 9.81 ms² downwards.

Sign Convention

As stated in the problem, the positive $y$-axis is chosen to be vertically upward. Since gravity acts downward, it will have a negative sign for the $y$-component of the acceleration for both the ascending and descending parts of the motion. Based on this analysis, the correct option is: d) negative, negative

Key concepts

Acceleration Due to Gravity

Understanding the acceleration due to gravity is crucial when studying projectile motion. This acceleration is a constant force that acts on all objects near the Earth's surface. It pulls objects toward the Earth's center at a standard rate of approximately $9.81{\text{ m/s}}^2$, and it is always directed downward, towards the ground.

When an object is projected into the air, it is this force that causes the object's vertical velocity to decrease as it rises and increase as it falls. This is because the acceleration due to gravity is working against the upward motion during the ascent, and it accelerates the object during the descent. Regardless of the direction of motion, the acceleration due to gravity is consistently pulling downward with the same magnitude.

Therefore, when using a coordinate system where up is positive, the acceleration due to gravity will have a negative value, which impacts our calculations and interpretations of projectile motion. This understanding is essential for identifying the correct direction and magnitude of an object's acceleration at any point in its trajectory.

Parabolic Trajectory

A parabolic trajectory is the curved path that a projectile follows under the influence of gravity alone, assuming no other forces like air resistance are acting on it. The shape of the path resembles a parabola, which is mathematically described as a curve where any point is at an equal distance from a fixed point (the focus) and a fixed straight line (the directrix).

In projectile motion, the parabolic trajectory is the result of the horizontal velocity component remaining constant while the vertical velocity component is being altered by gravity. The beauty of this motion is that it can be predicted using basic physics principles. For any given initial velocity, angle of launch, and ignoring air resistance, the path of the projectile can be precisely calculated.

The arc formed by a projectile, such as a basketball shot towards a hoop or a cannonball fired from a cannon, can be broken down into its horizontal and vertical motions and analyzed to predict landing positions and optimize angles for maximum range or height. This understanding of parabolic trajectory is not only foundational in physics but also has practical applications in sports, engineering, and any field that involves the motion of objects through the air.

Vertical and Horizontal Velocity Components

In projectile motion, understanding the vertical and horizontal velocity components are essential for predicting the movement of the projectile. The horizontal component of velocity, $v_x$, and the vertical component, $v_y$, work independently of each other.

The horizontal velocity component remains constant throughout the motion since, in an ideal scenario, no horizontal forces like air resistance are acting on the projectile. This is why an object projected horizontally will continue to move at the same speed until it hits the ground, assuming it doesn't encounter any resistance.

Conversely, the vertical velocity component is significantly influenced by gravity. At the start of the projectile's flight, this component is determined by the initial launch angle and speed. As the projectile rises, the vertical component of velocity decreases due to gravity's pull until it momentarily reaches zero at the projectile's peak. Then, as the projectile descends, the vertical component increases in the negative direction since it's accelerating downward.

A thorough comprehension of these components allows us to break down complex projectile motions into simpler calculations by focusing on horizontal and vertical motions separately. This approach simplifies the analysis and helps students understand the motion in a more manageable way. To analyze projectiles, we often use vector decomposition, where the velocity of an object is split into $v_x$ and $v_y$ and treated separately, yet together they define the projectile's flight.