Question

at what points are the vertical speeds of the ball the same value?
a points a and b
b points b and c
c points a and d
d points a, b, and c.
e all of the points.

Explanation:

Step1: Analyze vertical motion in projectile

In projectile motion, the vertical acceleration \(a = -g\) (constant). The vertical velocity \(v_y=v_{0y}-gt\). For two points to have the same vertical speed (magnitude of \(v_y\)), they must be at the same height (symmetric about the peak in some cases).

Step2: Check each option

• Option A: Points A and B are at different heights. Using \(v_y^2 - v_{0y}^2=2a\Delta y\), since \(\Delta y

eq0\), \(|v_{yA}|
eq|v_{yB}|\).

• Option B: Points B and C are at different heights. Using \(v_y^2 - v_{0y}^2=2a\Delta y\), since \(\Delta y

eq0\), \(|v_{yB}|
eq|v_{yC}|\).

• Option C: Points A and D. The vertical motion is symmetric. At the same height (if we consider the trajectory), the magnitude of vertical velocity is the same. Using \(v_y^2 - v_{0y}^2=2a\Delta y\), when \(\Delta y = 0\) (for same - height points in projectile, here A and D can be considered as same - height in a sense of vertical - speed symmetry), \(|v_{yA}| = |v_{yD}|\).
• Option D: Points A, B and C are at different heights. Using \(v_y^2 - v_{0y}^2=2a\Delta y\), since \(\Delta y

eq0\) for pairs among them, \(|v_{yA}|
eq|v_{yB}|
eq|v_{yC}|\).

• Option E: All points are at different heights (except maybe symmetric pairs). Using \(v_y^2 - v_{0y}^2=2a\Delta y\), since \(\Delta y

eq0\) for non - symmetric pairs, \(|v_{y}|\) are not all equal.

Answer:

C. Points A and D