91Ó°ÊÓ

In projectile motion, the conservation of momentum is a crucial concept. It states that in an isolated system, momentum remains constant unless acted on by an external force. Here, when the athlete throws the halteres, the momentum of the entire system before the throw is equal to the momentum after the throw.

Initially, the total system is the athlete plus the halteres, moving horizontally with a velocity of 9.5 m/s. This gives us an initial momentum, calculated by multiplying total mass by velocity: \(78 \times 9.5\).

When the athlete throws the halteres backward, they exert an equal and opposite force on him, altering his horizontal velocity. To find this new velocity, we apply the conservation of momentum equation: \(78 \times 9.5 = 67 \times v'_{x}\), where 67 kg is the athlete's mass without the halteres. Solving for \(v'_{x}\) determines how the athlete's speed changes, illustrating how momentum is conserved.

Horizontal velocity is a key aspect of projectile motion. It determines how far something travels horizontally, assuming no air resistance affects the motion.

In our scenario, the athlete starts with a horizontal velocity of 9.5 m/s. After throwing the halteres, his horizontal speed increases to approximately 11.05 m/s. This change happens due to the transfer of momentum, as explained earlier. The speed increase leads to the athlete covering a larger horizontal distance during his jump.

The horizontal velocity remains constant during flight since no horizontal forces (like friction or air resistance) are acting. Therefore, once the new velocity is established, it won't change until the athlete lands.

The range of a projectile is the horizontal distance traveled while in motion. To calculate range, you need to know both the horizontal velocity and the time of flight.

For the athlete, the improvement in range due to the halteres is calculated as:

• Initial horizontal velocity: 9.5 m/s
• New horizontal velocity after throwing: 11.05 m/s

The additional distance is computed by subtracting the initial range from the new range:

• Difference in velocity: \(11.05 - 9.5 = 1.55\) m/s
• Time of flight: approximately 0.816 s

Thus, the extra range achieved is approximately \(1.55 \times 0.816 \approx 1.26\) meters. This additional distance reflects the power of optimizing horizontal velocity in projectile motion through conservation of momentum.

The time of flight is how long a projectile remains in the air. It is influenced by the vertical motion components, particularly the vertical velocity and gravity.

Initially, the athlete's vertical launch velocity is 4.0 m/s. The time to reach peak height is determined by the formula \( t_{up} = \frac{v_y}{g} \), where \( g \) is the acceleration due to gravity (9.8 m/s²). The calculated time to ascend and descend is doubled to find the total flight time, given as \( t = \frac{2 \times 4.0}{9.8} \approx 0.816 \) seconds.

This time of flight remains constant whether or not the halteres are used, as they do not affect vertical motion. Hence, the increased horizontal range comes purely from heightened horizontal velocity, not extending the time in the air.