Ch 02: Motion Along a Straight Line

Young & Freedman Calc - University Physics 15th Edition

Young & Freedman Calc15th EditionUniversity PhysicsISBN: 9780135159552Not the one you use?Change textbook

Young & Freedman Calc 15th Edition

Ch 02: Motion Along a Straight Line

Problem 30

Chapter 2, Problem 30

A small block has constant acceleration as it slides down a frictionless incline. The block is released from rest at the top of the incline, and its speed after it has traveled $6.80$ m to the bottom of the incline is $3.80$ m/s. What is the speed of the block when it is $3.40$ m from the top of the incline?

Verified step by step guidance

Identify the known values: initial speed (u) = 0 m/s (since the block is released from rest), final speed (v) = 3.80 m/s after traveling a distance (s) = 6.80 m.

Use the kinematic equation for constant acceleration: $ v^2 = u^2 + 2as $ to find the acceleration (a). Substitute the known values: $ (3.80)^2 = 0 + 2a(6.80) $. Solve for a.

Once the acceleration is found, use it to determine the speed of the block when it is 3.40 m from the top. Use the same kinematic equation: $ v^2 = u^2 + 2as $, where s = 3.40 m.

Substitute the known values into the equation: $ v^2 = 0 + 2a(3.40) $.

Solve for the new speed (v) when the block has traveled 3.40 m down the incline.

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Key Concepts

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

Kinematics

Kinematics is the branch of physics that deals with the motion of objects without considering the forces that cause the motion. It involves equations that relate displacement, velocity, acceleration, and time. In this problem, kinematic equations help determine the block's speed at different points along the incline.

Constant Acceleration

Constant acceleration implies that the rate of change of velocity is uniform over time. This simplifies calculations as the kinematic equations can be applied directly. For the block on the incline, knowing it has constant acceleration allows us to use these equations to find its speed at any point along the path.

Energy Conservation

Energy conservation in physics states that the total energy in a closed system remains constant. For a frictionless incline, mechanical energy (potential and kinetic) is conserved. As the block slides down, potential energy converts to kinetic energy, allowing us to calculate its speed at different positions using energy principles.

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Related Practice

Textbook Question

A cat walks in a straight line, which we shall call the $x$ -axis, with the positive direction to the right. As an observant physicist, you make measurements of this cat's motion and construct a graph of the feline's velocity as a function of time (Fig. E $2.30$ ). What distance does the cat move during the first $4.5$ s? From $t equals 0$ to $t equals 7.5$ s?

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Textbook Question

If a flea can jump straight up to a height of $0.440$ m, what is its initial speed as it leaves the ground?

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Textbook Question

A cat walks in a straight line, which we shall call the $x$ -axis, with the positive direction to the right. As an observant physicist, you make measurements of this cat's motion and construct a graph of the feline's velocity as a function of time (Fig. E $2.30$ ). What is the cat's acceleration at $t equals 3.0$ s? At $t equals 6.0$ s? At $t equals 7.0$ s?

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Textbook Question

If a flea can jump straight up to a height of $0.440$ m, How long is it in the air?

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Textbook Question

A small rock is thrown vertically upward with a speed of $22.0$ m/s from the edge of the roof of a $30.0$ -m-tall building. The rock doesn't hit the building on its way back down and lands on the street below. Ignore air resistance. What is the speed of the rock just before it hits the street?

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Textbook Question

A cat walks in a straight line, which we shall call the $x$ -axis, with the positive direction to the right. As an observant physicist, you make measurements of this cat's motion and construct a graph of the feline's velocity as a function of time (Fig. E $2.30$ ). Assuming that the cat started at the origin, sketch clear graphs of the cat's acceleration and position as functions of time.

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