Question

A ball is dropped from a bridge \(125 \mathrm{~m}\) above a river. 2s later a second ball is thrown straight down. What must be the initial velocity of second ball so that both hit the water surface simultaneously? (1) \(\frac{80}{3} \mathrm{~m} / \mathrm{s}\) (2) \(\frac{160}{3} \mathrm{~m} / \mathrm{s}\) (3) \(\frac{40}{3} \mathrm{~m} / \mathrm{s}\) (4) \(\frac{20}{3} \mathrm{~m} / \mathrm{s}\)

Short answer

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Step-by-step solution

Understanding the problem

A ball is dropped from a bridge 125 meters above a river. After 2 seconds, a second ball is thrown straight down. We need to find the initial velocity of the second ball so that both balls hit the water simultaneously.

Determine the time it takes the first ball to reach the water

Use the kinematic equation for an object under constant acceleration (gravity): . . . So, solving for .

Calculate the time

Rearranging the equation to solve for time: . So, . .

Time for the second ball to reach the water

Since the first ball takes to hit the water and is dropped 2 seconds before the second ball, the second ball will have .

Find the initial velocity of the second ball

Using the kinematic equation: . We need to solve for v_0 . = displacement = = time = , and solving for initial velocity gives .

Key concepts

Kinematic Equations

Kinematic equations are essential for solving problems involving motion with constant acceleration, like our exercise. These equations relate the displacement, initial velocity, final velocity, acceleration, and time. For the motion of an object under gravity, like a falling ball, the most frequently used equation is:



Here,
used is:

Understanding Variables:

• \( s \): displacement
• \( v_0 \): initial velocity
• \( t \): time
• \( a \): acceleration

In our exercise, we use this formula to find out how long it takes for objects to fall and their initial velocities. By rearranging the equation, we can solve for the unknown variable. When you understand how to manipulate these variables, you can solve any kinematic problem.

Constant Acceleration

Constant acceleration means that an object's velocity changes by a constant amount every second. For objects in free fall, like the balls in our problem, this acceleration is due to gravity, denoted by \( g \)

In our kinematic equations, we typically use \( g \) as \( 9.8 \space m/s^2 \). When solving these problems, it’s vital to know:

• Acceleration is consistent, so every second, the speed of the object increases by 9.8\(\space m/s\).
• In free fall, initial velocity (\( v_0 \)) and acceleration (\( g \)) are what mainly affect the motion.

For example, without considering air resistance, an initially stationary ball dropped from a height will speed up uniformly at 9.8\(\space m/s\) every second. If thrown downwards, its initial velocity gets added to the speed increment due to gravity. Hence, understanding and calculating the impacts of this constant acceleration are fundamental for solving such kinematic problems.

Initial Velocity Calculations

Initial velocity (\( v_0 \)) is the speed at which an object begins its motion. It’s an essential part of kinematic problems because it sets up the entire motion sequence. For our exercise, let's break down the steps to find out the initial velocity of the second ball:

We need to know:

• The height from which the ball is dropped (125 meters).
• The time it takes for each ball to hit the ground.
• The time difference between the release of the two balls.

For the first ball, dropped with \( v_0 = 0 \), we use:

Solving for time (:

So, the first ball takes approximately 5 seconds to hit the water. Since the second ball is thrown 2 seconds later, it has only 3 seconds to reach the water. We need the initial velocity ( Taking displacement (s) as -125 meters ,-125metersWhere gravity (g) is -9.8m/s ^2time ( t) is 3 seconds gives Re-arranging the formula results inSet up properly, initial velocity can determine the entire motion pathway of a projectile. So mastering these initial velocity calculations is crucial for any kinematic exercise.