A coin is placed near the edge of a table and is flicked horizontally. It leaves the table with a
horizontal velocity of 25 cm/s. The height of the table is 1.225m. Find the time taken by the coin to reach the ground.
Answers
Before I answer this question, let me explain something: First, this question can be resolved in the realm of Newtonian physics. The force of gravity acts the same on every object regardless of size or shape or mass. The force is directly proportional to the mass of the object and the local gravitational constant (acceleration) and nothing else of any consequence. This means that all objects dropped to the surface of the earth will accelerate at the same rate, unless an outside force acts on it to modify the fall. In the case of dropping a feather or a cannon ball, in a vacuum they will fall at the same rate and hit the earth at the same time if dropped from the same height at the same instant of time. If the experiment is done in air, the large surface area to mass ratio of the feather means that the air density will have a significant effect on the rate at which the feather falls. It reaches terminal velocity very quickly and flutter slowly to the ground, whereas the cannot ball with a much smaller surface area to mass ratio does not. Terminal velocity occurs when the air drag on a falling object equals the downward force of gravity on that object. Now for your answer.
You have described a moving quarter that strikes a dime that is hanging over the edge of the table. For this question we can ignore the air drag problem, since the falling objects from the table to the floor will not gather much speed, and the two objects have similar mass densities and are of similar shapes, the air drag will tend to affect each of them approximately the same. Now what occurs is that the quarter is accelerated to a speed that gives it both momentum and energy at the point of release. The energy (= 0.5Mv²) and momentum (= Mv) is imparted to the quarter by someone’s finger upon flicking the quarter toward the dime. When it strikes the dime, the laws of conservation of momentum and energy govern, and the dime gains some of both momentum and energy from that of the quarter, and then flies off of the table and begins falling to the ground. Since both momentum and energy are conserved if the impact between quarter and dime is an efficient process, the dime takes on velocity v1 and the quarter slows down from v to v2 in such a manner that the sums of both energy and momentum of the two objects is conserved, that is
For momentum: mv1 + Mv2 = Mv
Where M = mass of quarter, m = mass of dime, v1 = velocity of dime after impact, v2 = velocity of quarter after impact, and v = velocity of quarter just before impact.
For Energy: 0.5(m v1² + M v2²) = 0.5 M v²
The correct answers for v1 and v2 are found when both of these equations are satisfied. I leave it to the reader to solve these algebraic simultneous equations.
Still, this does not address your question of “Which one hits the floor first, and why do they hit at the same time?” Now you should note that all of the above will allow you to calculate what you need to know to solve the question of relative speeds of the two coins, but I put this in as a means to understand the physics at play, but even with all the hoopla above, this does not address your problem directly.
