Block A is traveling with a speed v0 on a smooth surface...

Block $A$ is traveling with a speed $v_{0}$ on a smooth surface when the surface suddenly becomes rough with a coefficient of friction of $\mu$ causing the block to stop after a distance $d$. If block $A$ were traveling twice as fast, that is, at a speed $2 \%$, how far will it travel on the rough surface before stopping? a. $d / 2$ b. $d$ c. $\sqrt{2} d$ d. $2 d$ e. $4 d$

Block $A$ is traveling with a speed $v_{0}$ on a smooth surface when the surface suddenly becomes rough with a coefficient of friction of $\mu$ causing the block to stop after a distance $d$. If block $A$ were traveling twice as fast, that is, at a speed $2 \%$, how far will it travel on the rough surface before stopping?
a. $d / 2$
b. $d$
c. $\sqrt{2} d$
d. $2 d$
e. $4 d$

Key Concepts

Quadratic Relationship between Speed and Stopping Distance

Because kinetic energy increases with the square of the speed, the stopping distance on a rough surface (assuming constant friction) is also proportional to the square of the initial speed. This means that if the speed is doubled, the stopping distance increases by a factor of four, highlighting the quadratic dependence of stopping distance on speed.

Friction and Coefficient of Friction

Friction is a resistive force that acts opposite to the direction of motion. The coefficient of friction (?) quantifies the roughness of the surface and, when multiplied by the normal force, gives the magnitude of the frictional force. This force is responsible for doing work on the moving object, thereby dissipating its kinetic energy and eventually stopping it.

Work-Energy Principle

This principle states that the work done by forces acting on an object is equal to the change in its kinetic energy. In the context of motion on a rough surface, the work done by frictional forces brings the object to a stop by dissipating its kinetic energy.

Kinetic Energy

Kinetic energy is the energy possessed by an object due to its motion and is given by the expression (1/2)mv². The quadratic relationship between velocity and kinetic energy means that if the speed is doubled, the kinetic energy increases by a factor of four, which has direct implications on the distance required to bring the object to a stop under friction.

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