Problem 45 - University Physics with Modern Physics

Question

A sly 1.5-kg monkey and a jungle veterinarian with a blow-gun loaded with a tranquilizer dart are 25 m above the ground in trees 70 m apart. Just as the veterinarian shoots horizontally at the monkey, the monkey drops from the tree in a vain attempt to escape being hit. What must the minimum muzzle velocity of the dart be for the dart to hit the monkey before the monkey reaches the ground?

Step-by-Step Solution

Answer

The minimum muzzle velocity is approximately 31 m/s.

Step 1: Understand the Scenario

We have a monkey and a veterinarian both at 25 meters above the ground. The monkey drops from the tree at the same moment the veterinarian fires a dart horizontally at the monkey. The horizontal distance between them is 70 meters. We need to find the minimum muzzle velocity of the dart that will still hit the monkey before it hits the ground.

Step 2: Analyze Vertical Motion

Both the monkey and the dart fall freely under gravity with the same vertical acceleration. The time it takes for the monkey to hit the ground can be calculated with the equation of motion: \[ h = \frac{1}{2} g t^2 \]where \( h = 25 \) meters and \( g = 9.8 \text{ m/s}^2 \). Solve for \( t \).

Step 3: Calculate Time of Fall

Rearranging the previous equation, we have:\[ t^2 = \frac{2h}{g} = \frac{2 \times 25}{9.8} \approx 5.10 \]So,\[ t = \sqrt{5.10} \approx 2.26 \text{ s} \]This is the time both the dart and the monkey are in free fall before reaching the ground.

Step 4: Determine Minimum Muzzle Velocity

The dart's horizontal motion must cover the 70-meter gap in the same time it takes to fall. Thus, the horizontal velocity \( v_x \) is given by:\[ v_x = \frac{d}{t} \]where \( d = 70 \) meters and \( t = 2.26 \) seconds:\[ v_x = \frac{70}{2.26} \approx 30.97 \text{ m/s} \].

Step 5: Conclusion

The minimum muzzle velocity for the dart to hit the monkey before it reaches the ground is approximately \( 31 \text{ m/s} \).

Key Concepts

Horizontal Velocity

Understanding horizontal velocity helps us solve problems like the monkey and dart scenario in projectile motion challenges. Here, the dart is fired horizontally towards the monkey immediately. This horizontal velocity, denoted as \( v_x \), is crucial for determining whether the dart can cover the horizontal gap of 70 meters in the same time it takes for the objects to fall. The movement is purely horizontal, meaning gravity doesn't directly influence \( v_x \).

To compute this, we use the formula:

• \( v_x = \frac{d}{t} \)

Here, \( d \) is the horizontal distance to be traveled, which is 70 meters, and \( t \) is the time calculated based on vertical motion. The key takeaway is that horizontal velocity remains constant since there are no external horizontal forces acting within an ideal environment.

Vertical Motion

The concept of vertical motion in projectile exercises like this one involves understanding how objects fall under gravity's influence. Both the monkey and the dart experience free-fall motion vertically. Even if the dart is shot horizontally, it still begins to fall simultaneously with the monkey.

To find out how long both will fall before hitting the ground, we apply the basic equation of vertical motion:

• \( h = \frac{1}{2} g t^2 \)

Where \( h \) is the height (25 meters), \( g \) is the acceleration due to gravity (approximately \( 9.8 \text{ m/s}^2 \)), and \( t \) is the time we want to determine. Solving this gives us insight into how long their vertical journey will take and is key for calculating the necessary horizontal velocity for the dart.

Free Fall

In physics, free fall describes what happens when the only force acting on an object is gravity. In our exercise, both the monkey and the dart experience free fall from the moment they're subjected to gravity's downward force. This means they'll have the same vertical acceleration \( g = 9.8 \text{ m/s}^2 \).

Free fall ensures that the vertical aspect of their motion is identical, simplifying our calculations. With free fall:

• Each object, regardless of horizontal speed, falls at the same rate.
• Time to hit the ground is the same for both objects.

Knowing this allows us to ignore any vertical motion differences when determining if the dart will hit the monkey, focusing solely on horizontal distance.

Equations of Motion

Equations of motion provide the mathematical tools to describe the movement of objects through time under certain conditions, such as uniform acceleration, seen in our projectile problem. These equations include both horizontal and vertical components of motion separately.

Vertical motion is governed by:

• \( h = \frac{1}{2} g t^2 \)

Where we calculate the time of fall. For horizontal motion, we rely on:

• \( v_x = \frac{d}{t} \)

In our exercise, by solving these equations, we determine the necessary horizontal velocity that ensures the dart reaches the monkey before both land. Understanding these equations is vital for breaking down each aspect of motion in similar physics scenarios.