Question

The initial velocity of a projectile has a horizontal component equal to \(5 \mathrm{~m} / \mathrm{s}\) and a vertical component equal to \(6 \mathrm{~m} / \mathrm{s}\). At the highest point of the projectile's flight, what is (a) the horizontal component of its velocity and (b) the vertical component of its velocity? Explain.

Short answer

(a) Horizontal: 5 m/s, (b) Vertical: 0 m/s.

Step-by-step solution

Understanding the Problem

We need to find the horizontal and vertical components of the velocity of a projectile at its peak. Given that the horizontal component of the initial velocity is \(5 \,\mathrm{m/s}\) and the vertical component is \(6 \,\mathrm{m/s}\), we will determine these components at the highest point of its trajectory.

Analyze Horizontal Component of Velocity

In projectile motion, the horizontal component of the velocity remains constant throughout the flight. Since the initial horizontal velocity is \(5 \,\mathrm{m/s}\), it remains \(5 \,\mathrm{m/s}\) at the highest point.

Analyze Vertical Component of Velocity

At the highest point of trajectory, the vertical component of the projectile's velocity is \(0 \,\mathrm{m/s}\). This is because gravity momentarily halts the upward motion before making the projectile descend.

Key concepts

Horizontal Component of Velocity

In projectile motion, the horizontal component of velocity is crucial. It stays constant throughout the flight of the projectile.
The main reason behind this constancy is the lack of any horizontal acceleration (assuming air resistance is negligible). The initial horizontal velocity, provided as 5 m/s in the exercise, remains unchanged as the projectile moves.

• There are no forces acting horizontally to alter this component.
• It means the projectile maintains its horizontal speed from start to finish.

In practice, changes in the horizontal component can occur due to air resistance, but this is typically ignored in basic physics problems to simplify the study of motion.

Vertical Component of Velocity

The vertical component of velocity in projectile motion behaves differently from the horizontal one due to the influence of gravity.
It constantly changes because gravity affects only the vertical aspect of the motion. In our problem, the initial vertical velocity is 6 m/s.

• As the projectile ascends, gravity slows this velocity down.
• At the highest point of the trajectory, the vertical velocity becomes 0 m/s.

This is because gravity has completely counteracted the upward motion at that point, preparing the projectile to descend.

Trajectory

The path that the projectile follows through the air is known as the trajectory. It is typically a curved path, shaped like a parabola in projectile motion problems without air resistance.
This curve arises because while the horizontal component of velocity remains constant, the vertical component is constantly changing due to gravity.

• The trajectory is symmetrical, meaning the path up equals the path down, starting and ending at the same height.
• Understanding the trajectory helps predict where a projectile will land.

Therefore, it offers a visual representation of how both components of velocity work together to determine the projectile's motion.

Gravity

Gravity plays a crucial role in projectile motion, primarily affecting the vertical component of velocity. It continuously accelerates the projectile downwards at approximately 9.81 m/s².
At the peak of the projectile's flight, gravity is the reason why the vertical velocity becomes zero, halting upward motion.

• Gravity turns the upward motion into downward motion, ensuring the projectile returns to the ground.
• Without gravity, a projectile would continue indefinitely in the direction of its initial velocity.

Understanding gravity's influence is vital as it ties together the changes in motion and velocity for objects in projectile flight.