Question

Two divers run horizontally off the edge of a low cliff. Diver 2 runs with twice the speed of diver 1. (a) When the divers hit the water, is the horizontal distance covered by diver 2 twice as much as, four times as much as, or equal to the horizontal distance covered by diver 1? (b) Choose the best explanation from among the following: A. The drop time is the same for both divers. B. Drop distance depends on \(t^{2}\). C. All divers in free fall cover the same distance.

Short answer

(a) The horizontal distance is twice. (b) A. The drop time is the same for both divers.

Step-by-step solution

Understanding the Problem

Two divers are running off a low cliff, with Diver 2 running at twice the speed of Diver 1. We are to determine the horizontal distances they cover before hitting the water and find whether it is two times, four times, or the same for Diver 2 compared to Diver 1.

Horizontal Motion Analysis

For horizontal motion, the horizontal distance covered by an object is calculated as: \[ d = vt \] where \(v\) is the horizontal velocity and \(t\) is the time in the air. Since Diver 1 has velocity \(v\) and Diver 2 has velocity \(2v\), we will calculate this distance for both.

Time of Flight Calculation

The time both divers spend in the air is the same because the vertical motion is identical (starting from the same height and thus having the same vertical drop). This is supported by gravity acting equally regardless of horizontal velocity, making the drop time \(t\) identical for both divers.

Calculating Horizontal Distances

The horizontal distance for Diver 1 is \(d_1 = v \cdot t\). For Diver 2, the distance is \(d_2 = 2v \cdot t\). By inserting the time \(t\), which is the same for both, we get:- Diver 1: \(d_1 = vt\)- Diver 2: \(d_2 = 2vt\)

Conclusion on Distance Comparison

Comparing \(d_1\) and \(d_2\), we see that \(d_2 = 2 \times d_1\). Therefore, Diver 2 covers twice the horizontal distance of Diver 1.

Choosing Explanation

The correct explanation is "A. The drop time is the same for both divers." because both are affected by gravity alone after leaving the edge, meaning they fall for the same duration despite different horizontal speeds.

Key concepts

Horizontal Distance

When we're talking about horizontal distance in projectile motion, we are referring to how far an object travels across the ground. Imagine if you threw a ball parallel to the ground—how far it rolls is considered the horizontal distance.

• This distance can be calculated using the formula \[d = vt\]
• Here, \(v\) is the horizontal speed (or velocity) of the object, and \(t\) is the time the object is in the air.

In the case of the divers, Diver 1 runs with speed \(v\), while Diver 2 runs with \(2v\). Both divers start from the same point, so anything affecting them later on happens equally to both. Due to Diver 2 having twice the initial speed, they cover twice the horizontal distance when hitting the water as compared to Diver 1.

Gravity

Gravity is the invisible force that pulls objects towards the center of the Earth. It's what keeps us grounded and affects any object in free fall, such as the divers.

• Gravity pulls all objects downward with an acceleration of approximately \(9.8 \, \text{m/s}^2\).
• It acts equally on all objects, regardless of their weight or horizontal speed.

In projectile motion, gravity only affects the vertical component. For the divers, it ensures they both hit the water after the same time, since their vertical motion starts identical with the only varying factor being their initial horizontal velocity. Regardless of how fast Diver 2 runs horizontally, gravity's pull is the same for both, ensuring they fall for the same duration.

Time of Flight

The time of flight is the total time an object stays in the air before touching down. In projectile motion, this time depends on the initial height and gravitational pull, not on the horizontal speed.

• This is why two objects dropped from the same height will hit the ground at the same time, even if one is moving faster horizontally.
• In the divers' case, since both jump from the same height, they fall under the same conditions.Irrespective of diver speeds, both have the same time \(t\) in the air, as gravity impacts them equally.

This crucial understanding that time of flight is not related to horizontal speed helps explain why Diver 2 covers double the distance without taking longer to hit the water.

Vertical Motion

Vertical motion, in the context of projectile motion, refers to how an object moves up or down due to gravity. This type of motion can be calculated separately from the horizontal component, thanks to gravity acting downwards consistently.

• The forces causing vertical motion—gravity—is constant and at right angle to any initial horizontal motion.
• Given that both divers start their descent from the same height, the distance they fall is the same.

you can express vertical motion with the formula for distance, which is influenced by time squared, \(s = 0.5 \, g \, t^2\). The concept that their vertical drops are identical means while Diver 2 has twice the horizontal distance due to his speed, the path of their fall remains unchanged and equal.