Ch 02: Kinematics in One Dimension

Knight Calc - Physics for Scientists and Engineers 5th Edition

Knight Calc5th EditionPhysics for Scientists and EngineersISBN: 9780137344796Not the one you use?Change textbook

Knight Calc 5th Edition

Ch 02: Kinematics in One Dimension

Problem 57

Chapter 2, Problem 57

A lead ball is dropped into a lake from a diving board 5.0 m above the water. After entering the water, it sinks to the bottom with a constant velocity equal to the velocity with which it hit the water. The ball reaches the bottom 3.0 s after it is released. How deep is the lake?

Verified step by step guidance

Step 1: Calculate the velocity of the ball just before it hits the water. Use the kinematic equation for free fall:

v = \sqrt{2 g h}

, where

g

is the acceleration due to gravity (9.8 m/s²) and

h

is the height (5.0 m).

Step 2: Determine the time it takes for the ball to fall from the diving board to the water surface. Use the kinematic equation:

t = \sqrt{\frac{2}{h} / g}

, where

h

is the height (5.0 m) and

g

is the acceleration due to gravity.

Step 3: Subtract the time it takes to fall to the water surface (calculated in Step 2) from the total time (3.0 s) to find the time the ball spends sinking in the water.

Step 4: Use the constant velocity of the ball in the water (calculated in Step 1) and the time spent sinking (calculated in Step 3) to determine the depth of the lake. Use the formula:

d = v \cdot t

, where

v

is the velocity and

t

is the time spent sinking.

Step 5: Add the depth of the lake (calculated in Step 4) to the height of the diving board (5.0 m) to find the total depth from the diving board to the bottom of the lake.

Verified video answer for a similar problem:

This video solution was recommended by our tutors as helpful for the problem above.

Video duration:

Was this helpful?

Key Concepts

Here are the essential concepts you must grasp in order to answer the question correctly.

Free Fall and Gravitational Acceleration

When an object is dropped from a height, it undergoes free fall, accelerating downwards due to gravity at approximately 9.81 m/s². The distance fallen can be calculated using the equation d = 0.5 * g * t², where d is the distance, g is the gravitational acceleration, and t is the time of fall. This concept is essential for determining the velocity of the ball just before it hits the water.

Constant Velocity in Fluids

Once the lead ball enters the water, it sinks with a constant velocity, meaning it experiences no net acceleration. This occurs when the forces acting on it, such as buoyancy and drag, balance out the weight of the ball. Understanding this concept is crucial for calculating the time it takes for the ball to reach the bottom of the lake after entering the water.

Kinematic Equations

Kinematic equations describe the motion of objects under constant acceleration or constant velocity. In this scenario, we can use these equations to relate the distance traveled in the water to the time taken and the constant velocity. This allows us to find the total depth of the lake by adding the distance fallen before entering the water to the distance traveled while sinking.

Recommended video:

Guided course

08:25

Kinematics Equations

Related Practice

Textbook Question

Find an expression for the minimum stopping distance d_stop of a car traveling at speed v₀ if the driver's reaction time is T_react and the magnitude of the acceleration during maximum braking is a constant a_brake.

views

Textbook Question

You are 9.0 m from the door of your bus, behind the bus, when it pulls away with an acceleration of 1.0 m/s². You instantly start running toward the still-open door at 4.5 m/s. What is the maximum time you can wait before starting to run and still catch the bus?

views

Textbook Question

A 200 kg weather rocket is loaded with 100 kg of fuel and fired straight up. It accelerates upward at 30 m/s² for 30 s, then runs out of fuel. Ignore any air resistance effects. What is the rocket's maximum altitude?

views

Textbook Question

Ann and Carol are driving their cars along the same straight road. Carol is located at x = 2.4 mi at t = 0 h and drives at a steady 36 mph. Ann, who is traveling in the same direction, is located at x = 0.0 mi at t = 0.50 h and drives at a steady 50 mph. Draw a position-versus-time graph showing the motion of both Ann and Carol.

views

Textbook Question

A hotel elevator ascends 200 m with a maximum speed of 5.0 m/s. Its acceleration and deceleration both have a magnitude of 1.0 m/s². How long does it take to make the complete trip from bottom to top?

Textbook Question

A 1000 kg weather rocket is launched straight up. The rocket motor provides a constant acceleration for 16 s, then the motor stops. The rocket altitude 20 s after launch is 5100 m. You can ignore any effects of air resistance. What was the rocket's acceleration during the first 16 s?

views