Question

A firefighter jumps horizontally from a burning building with an initial speed of $1.5\frac{\mathrm{m}}{\mathrm{s}}.$ At what time is the angle between his velocity and acceleration vectors the greatest? (A) The instant he jumps (B) When he reaches terminal velocity (C) Halfway through his fall (D) Right before he lands on the ground

Short answer

The instant he jumps.

Step-by-step solution

- Understand the setup

The firefighter jumps horizontally with an initial speed of 1.5 m/s. The horizontal and vertical components of his motion need to be understood separately. The horizontal velocity remains constant at 1.5 m/s since there is no horizontal acceleration (assuming air resistance is negligible), while the vertical velocity increases due to gravity.

- Determine initial conditions

At the instant he jumps, the horizontal component of velocity is 1.5 m/s, and the vertical component of velocity is 0 m/s. The acceleration due to gravity acts vertically downward and has a magnitude of 9.8 m/s².

- Analyze the angle between velocity and acceleration

The angle between the velocity and acceleration vectors depends on the direction and magnitude of both vectors. Initially, the angle is 90 degrees because the horizontal velocity is 1.5 m/s and the vertical velocity is 0 m/s while acceleration is purely vertical at 9.8 m/s² downward. As he falls, the vertical component of velocity increases due to gravitational acceleration, changing the angle.

- Conclusion

From the initial condition, right when he jumps, the velocity vector points horizontally while the acceleration vector points vertically downwards. This configuration results in the maximum angle of 90 degrees between the two vectors.

Key concepts

Velocity and Acceleration Vectors

When dealing with projectile motion, it's crucial to grasp the concepts of velocity and acceleration vectors.
A vector has both magnitude and direction. In this scenario, the firefighter's initial velocity is entirely horizontal, with no vertical component. Therefore, the velocity vector points horizontally as he jumps.

Acceleration, however, always acts downward due to gravity. This gravitational acceleration is a constant force pulling objects towards the Earth at $\text{Missing left or extra right}$.
When discussing the angle between the velocity and acceleration vectors:

• Right as the firefighter jumps, the vertical acceleration due to gravity contrasts sharply with the purely horizontal velocity. This creates a 90-degree angle.
• As time progresses, the firefighter gains vertical velocity. This change modifies the angle between his velocity and the downward acceleration vector.

Horizontal and Vertical Components of Motion

Understanding horizontal and vertical components helps break down complex motions into more manageable parts.
Since the firefighter jumps horizontally with a speed of $\text{Extra close brace or missing open brace}$, his horizontal motion remains unaffected by gravity. This means:

• The horizontal velocity stays constant at $\text{Extra close brace or missing open brace}$.
• There's no horizontal acceleration (assuming no air resistance).

The vertical motion is a different story, driven entirely by gravitational acceleration:

• Initially, vertical velocity is zero right when he jumps.
• As he falls, this vertical velocity increases due to acceleration from gravity.
• Vertical velocity can be found using the equation $\text{Missing left or extra right}$, where $\text{Missing left or extra right}$ is gravitational acceleration and $\text{Missing left or extra right}$ is time.

The combined motion in both horizontal and vertical components determines the overall path, forming a parabolic trajectory typical of projectile motion.

Gravitational Acceleration

Gravity is a fundamental force influencing all objects near Earth.
In projectile motion, gravitational acceleration plays a crucial role. Gravitational acceleration has:

• A magnitude of $\text{Missing left or extra right}$.
• A direction pointing straight down towards the Earth's center.

This constant downward force impacts the vertical component of an object's motion, creating the curved paths typical of projectiles.
Some key points about gravitational acceleration:

• It doesn't affect horizontal motion if we ignore air resistance.
• It causes the vertical velocity of a falling object to increase, making the object accelerate downwards.

In summary, gravitational acceleration is vital for understanding how and why objects move the way they do when thrown or dropped, and it's precisely due to this force that the firefighter's vertical speed continually rises during his fall.