Problem 59 - College Physics

Question

\(\bullet\) In a volcanic eruption, a 2400 -kg boulder is thrown vertically upward into the air. At its highest point, it suddenly explodes (due to trapped gases) into two fragments, one being three times the mass of the other. The lighter fragment starts out with only horizontal velocity and lands 274 m directly north of the point of the explosion. Where will the other fragment land? Neglect any air resistance.

Step-by-Step Solution

Answer

The larger fragment will land 822 meters south of the explosion point.

Step 1: Understand the Problem

In the given problem, a boulder explodes into two fragments at its highest point. We need to determine where the heavier fragment lands. The explosion is symmetrical and we only know the position where the lighter fragment, which travels horizontally, lands.

Step 2: Use Momentum Conservation Principle

In the absence of external forces, the total momentum before and after the explosion is conserved. Initially, at the highest point of ascent, all velocity is vertical, so horizontal momentum is zero. Hence, the total horizontal momentum after explosion is also zero.

Step 3: Calculate Mass of Fragments

The total mass of the boulder is 2400 kg. Let the mass of the smaller fragment be \( m \). Therefore, the mass of the larger fragment will be \( 3m \). Given the equation \( m + 3m = 2400 \), we get \( 4m = 2400 \) so \( m = 600 \) kg. Thus, the masses are 600 kg for the smaller fragment and 1800 kg for the larger fragment.

Step 4: Apply Conservation of Momentum Horizontally

After the explosion, the momentum of this system remains balanced because the total initial horizontal velocity is zero. So, if the lighter fragment of mass 600 kg moves with velocity \( v \), the heavier fragment mass 1800 kg must move in the opposite direction with velocity \( -\frac{v}{3} \), keeping momentum balanced: \( 600v + 1800(-\frac{v}{3}) = 0 \).

Step 5: Analyze the Trajectories

The smaller fragment lands 274 meters north. Given that its horizontal motion is due to the explosion, the travel time \( t \) is the same for both fragments since both started from the highest point and free fall under gravity. Therefore, the smaller fragment lands north after traveling for \( t \) time, while the heavier fragment should land south at a different distance.

Step 6: Calculate Landing Distance of Larger Fragment

From momentum conservation, heavier fragment velocity equals \( -\frac{v}{3} \) (opposite direction of smaller fragment). If smaller fragment travels 274 m north, the larger must land \( -3 imes 274 \) m south as it travels for the same time \( t \). This gives the landing position as 822 m south.

Key Concepts

Explosion Physics

In explosion physics, when a boulder explodes due to trapped volcanic gases, the principle of momentum conservation becomes crucial. During the explosion, no external horizontal forces are acting on the system. This implies that the total horizontal momentum remains conserved. The explosion takes place at the boulder's highest point, meaning it initially has zero horizontal velocity.

• Prior to exploding, any upward velocity translates into potential energy.
• After the explosion, the sum of momenta of the fragments remains unchanged due to absence of external forces.

This conservation is pivotal in determining subsequent movement of the fragments post-explosion.

Vertical Motion

Vertical motion in this scenario refers to the initial throwing up and subsequent free fall of the boulder. The concept hinges on physics of projectiles without air resistance, where motions in horizontal and vertical directions are separate.

• The boulder ascends with kinetic energy translating into potential energy, reaching the maximum height where velocity is momentarily zero.
• The explosion occurs at this peak point, not affecting its vertical motion further.

For each fragment, the time to fall back to Earth remains same, as only vertical aspects like gravity influence this time frame.

Mass Distribution

Mass distribution is crucial when determining how fragments will behave following the explosion. Initially, the boulder is a single mass of 2400 kg, splitting into two.

• The smaller fragment, post-explosion, is 600 kg, while the larger, being three times this, is 1800 kg.
• The mass distribution directly influences the velocity each fragment acquires horizontally.

Mass significantly affects how each piece moves post-explosion, informing us how far and in what direction each fragment travels.

Trajectory Analysis

Trajectory analysis involves understanding the path taken by the fragments after the explosion. The horizontal displacement can be discerned using momentum conservation relationships.

• The smaller mass moves in one direction, while the larger moves oppositely due to mass and velocity balance.
• Given that the smaller fragment lands 274 meters north due to its acquired horizontal velocity, the larger fragment, by analysis, lands 822 meters south.

This analysis helps predict each fragment’s final location by integrating their initial trajectories and velocities.