Question

two balls are dropped at the same time from the top of a ladder. the balls are of equal size, but ball a has more mass than ball b. which of the following best describes the time it would take the two balls to hit the ground?
(note: ignore the effects of friction.)

a. ball a and ball b would hit the ground at the same time.
b. ball b would hit the ground before ball a.
c. there is not enough information given to answer the question.
d. ball a would hit the ground before ball b.

Explanation:

Identify the given physical conditions

We are given two balls of equal size dropped simultaneously from the same height (the top of a ladder). Ball A has more mass than Ball B (\(m_A > m_B\)). The problem explicitly states to ignore the effects of friction (air resistance).

Apply the equations of motion for free fall

In the absence of air resistance, the only force acting on each ball is gravity. According to Newton's second law of motion:

$$F_{\text{net}} = m \cdot a$$

Since gravity is the only force:

$$m \cdot g = m \cdot a$$

Dividing both sides by mass \(m\), we find that the acceleration \(a\) of any falling object is equal to the acceleration due to gravity \(g\):

$$a = g$$

Determine the time to hit the ground

The kinematic equation relating displacement \(d\), initial velocity \(v_i\), acceleration \(a\), and time \(t\) is:

$$d = v_i \cdot t + \frac{1}{2} a \cdot t^2$$

Since both balls are dropped from rest, the initial velocity \(v_i = 0\). The equation simplifies to:

$$d = \frac{1}{2} g \cdot t^2$$

Solving for time \(t\):

$$t = \sqrt{\frac{2d}{g}}$$

Compare the times for both balls

Because both balls are dropped from the same height \(d\) and experience the same gravitational acceleration \(g\), the time \(t\) depends only on \(d\) and \(g\). It is completely independent of the mass of the objects. Therefore, both Ball A and Ball B will take the same amount of time to hit the ground.

Answer:

• (A) Ball A and Ball B would hit the ground at the same time. (Correct answer)
• (B) Ball B would hit the ground before Ball A.
• (C) There is not enough information given to answer the question.
• (D) Ball A would hit the ground before Ball B.