Question

A hovering fly is approached by an enraged elephant charging at \(2.1 \mathrm{~m} / \mathrm{s}\). Assuming that the collision is elastic, at what speed does the fly rebound? Note that the projectile (the elephant) is much more massive than the target (the fly).

Short answer

The fly rebounds with a speed of approximately 4.2 m/s.

Step-by-step solution

Understand Elastic Collision

In an elastic collision, momentum (mass x velocity) and kinetic energy are conserved. Considering the great mass difference between the elephant and the fly, the momentum change of the elephant is negligible.

Conservation of Momentum

In the case where the mass of one object is much greater than the other, during elastic collision, the smaller object rebounds with nearly twice the velocity of the larger object. Here, because the mass of the elephant is much higher than the fly, we can use this fact to find the speed of the rebounding fly.

Calculate the Speed of the Rebounding Fly

So, the speed of the rebounding fly will be nearly twice the velocity of the elephant, which is \(2 \times 2.1 \, m/s = 4.2 \, m/s\). This is assuming the collision is perfectly elastic and that the elephant does not slow down upon impact.

Key concepts

Conservation of Momentum

In physics, the conservation of momentum is a fundamental principle, particularly relevant in collision scenarios. Momentum, which is the product of an object's mass and velocity, remains constant in an isolated system. This is true even when two objects collide, as long as no external forces act upon them.
For any two objects involved in a collision, such as an elephant and a fly, the total momentum before the collision is equal to the total momentum after the collision. This can be expressed as: \[ m_1v_1 + m_2v_2 = m_1v_1' + m_2v_2' \] Where:

• \(m_1, m_2\) are the masses of the two objects,
• \(v_1, v_2\) are their initial velocities,
• \(v_1', v_2'\) are their velocities after collision.

In cases of elastic collisions, like the one in this exercise, both momentum and kinetic energy are conserved. For an elephant (much more massive) colliding with a fly (less massive), the elephant's velocity effectively remains unchanged. This allows us to predict how fast the fly rebounds, focusing on the change in the fly’s velocity while treating the elephant as nearly constant.

Kinetic Energy

Another essential idea in elastic collisions is the conservation of kinetic energy. Kinetic energy, the energy that an object possesses due to its motion, is given by the formula: \[ KE = \frac{1}{2}mv^2 \]
For elastic collisions, the total kinetic energy before and after the collision remains unchanged. This means that even after the objects collide, the sum of their kinetic energies is the same.
Consider the situation with our elephant and fly. Here, the elephant carries a vast majority of the system's kinetic energy because of its large mass and relatively high velocity.
The equation for the conservation of kinetic energy reads as: \[ \frac{1}{2}m_1v_1^2 + \frac{1}{2}m_2v_2^2 = \frac{1}{2}m_1v_1'^2 + \frac{1}{2}m_2v_2'^2 \] Since the elephant's kinetic energy is dominant and mostly unaffected, the smaller fly's rebound speed increases to maintain the total system's energy equilibrium.

Mass Difference in Collisions

In a collision, the difference in mass between two colliding objects plays a crucial role in determining their post-collision velocities. If an object with a very large mass collides with one of a much smaller mass, like an elephant with a fly, interesting outcomes occur.
Due to the considerable mass difference, the more massive object (elephant) retains most of its initial speed. The smaller object (fly), however, reverses direction almost instantaneously at nearly twice the speed of the more massive one. This is a direct result of both the conservation of momentum and the conservation of kinetic energy.
In elastic collision scenarios where one mass is much larger, you can often assume that the larger mass does not significantly change its velocity. This simplified analysis is essential in predicting outcomes, such as the speed of the smaller object's rebound, resulting from their collision.
In summary, understanding mass difference is crucial in analyzing elastic collisions, enabling predictions about velocity changes and directions post-collision.