Question

cecilia dropped a ball from a height of 2 m. the ball fell in a straight path down to the ground. according to newtons universal law of gravitation, which would change the path of the ball so it traces out the widest curved path possible?

o tossing it horizontally at a slower speed
o tossing it horizontally at a faster speed
o dropping it from a shorter distance
o dropping it from a greater distance

Explanation:

When a ball is tossed horizontally, its horizontal motion and vertical motion (due to gravity) are independent. The horizontal displacement \(x = v_{0x}t\) (where \(v_{0x}\) is the initial horizontal speed and \(t\) is the time of flight). The time of flight \(t\) for a ball dropped from height \(h\) (vertical motion \(h=\frac{1}{2}gt^{2}\), so \(t = \sqrt{\frac{2h}{g}}\)) is determined by the height. If we toss the ball horizontally at a faster speed, with the same time of flight (since the height is fixed for a given drop - height scenario, assuming we are comparing tosses from the same height), a larger \(v_{0x}\) will lead to a larger horizontal displacement. A larger horizontal displacement for the same vertical displacement (determined by the height) will result in a wider - curved path (parabolic path \(y = h-\frac{1}{2}g(\frac{x}{v_{0x}})^{2}\)).

• Tossing it horizontally at a slower speed: Will give a smaller horizontal displacement for the same time of flight (from the fixed height), so a less - wide curved path.
• Dropping it from a shorter distance: Changes the time of flight (\(t=\sqrt{\frac{2h}{g}}\), smaller \(h\) means smaller \(t\)), but if we want a wide - curved path (more horizontal spread for a given vertical drop), a shorter drop height is not the way.
• Dropping it from a greater distance: Increases the time of flight. But if we consider the ratio of horizontal to vertical displacement (to get a “wide - curved” path in terms of the shape of the parabola), for a given horizontal speed \(v_{0x}\), \(y = h-\frac{1}{2}g(\frac{x}{v_{0x}})^{2}\). A higher \(h\) will make the parabola open up more in the vertical direction, but we are interested in the horizontal spread for a more “wide - curved” (in the sense of horizontal - to - vertical ratio) path when compared to horizontal tosses.

Answer:

tossing it horizontally at a faster speed