One ball is thrown horizontally while another is released from rest at the same height. Their motion is resolved into horizontal and vertical components to determine the time each takes to reach the floor.
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高さ4.9 mから一方を横に投げ、他方を同時に放す。先に床に着くのは？
空気抵抗を受けるピンポン球では、先に床に着くのは？
The equations explain the phenomenon
1Position is represented by an arrowr=(x, y)……
2Velocity is obtained from the change in positionv=drdt=(dxdt, dydt)……
3Acceleration is obtained from the change in velocitya=dvdt=(dvxdt, dvydt)……
4The equation of motion is resolved into horizontal and vertical componentsm dvxdt=0，m dvydt=−mg……
5The velocity of a ball thrown horizontally is calculatedvx=v₀，vy=−g t……
6The position of a ball thrown horizontally is calculatedx=v₀ t，y=h − ½ g t²……
7The landing time for a horizontal launch is calculatedt=√(2hg)……
The next sections examine motion with air resistance and launches at an angle.
8Air resistance is includedm dvdt=m (0, −g)−k |v| v……
9Air resistance is resolved into horizontal and vertical componentsm dvydt=−m g−k √(vx² + vy²) vy……
10The position of a ball launched at an angle is calculated without air resistancex=v₀ cosθ t，y=v₀ sinθ t − ½ g t²……
11Time t is eliminated to obtain the trajectory equationy=tanθ x−g x²2 v₀² cos²θ……
12The horizontal range on returning to the launch height is calculatedR=v₀² sin2θg……
13The angle for maximum range with air resistance is calculatedθmaximum range≒39.0°＜45°……
What this chapter has shown
Without air resistance, the horizontal speed does not appear in the vertical equation of motion. A ball thrown horizontally and a ball dropped simultaneously from the same height reach the floor at the same time if both have an initial vertical velocity of 0.
1.000 timesThe ratio of times without air resistance
In this air resistance model, the horizontal speed affects the vertical motion, and the thrown ball arrives 0.070 seconds later.
In everyday lifeAn eraser that leaves the edge of a desk almost horizontally and an eraser released from rest simultaneously at the same height reach the floor at nearly the same time if air resistance is small.
Detailed summary
When position, velocity, and acceleration are represented by arrows, the equation of motion separates into 2 equations, one horizontal and one vertical. For a thrown ball, the horizontal force is 0, so the horizontal velocity is constant, while the vertical motion follows the same equations as a ball thrown straight upward. Together, these motions produce a parabolic path. The horizontal speed does not appear in the vertical equation, so a ball thrown horizontally and a ball dropped from the same height reach the floor at the same time. With air resistance, the horizontal speed enters the vertical equation through |v|, changing both this result and the conclusion that a launch angle of 45° gives the greatest range.