Q: A block of mass $m$ is projected up a rough incline (angle $\theta$) with initial speed $v_0$. The coefficient of kinetic friction is $\mu_k$. What is the maximum distance $d$ the block travels up the incline before coming to rest?

$d = \frac{v_0^2}{2g(\sin\theta + \mu_k \cos\theta)}$

Question 1

A particle of mass $m$ slides down a frictionless incline of length $L$ at an angle $	heta$ to the horizontal. If the particle starts from rest, what is its speed when it reaches the bottom of the incline?

Accepted Answer

$v = \sqrt{2gL\sin	heta}$

Question 2

A box of mass $m$ is pushed up a rough incline (angle $	heta$) with a constant force $F$ parallel to the incline. The coefficient of kinetic friction is $\mu_k$. What is the work done by friction as the box moves a distance $d$ up the incline?

Question 3

A block of mass $m$ is compressed against a spring (spring constant $k$) by a distance $x_0$ on a frictionless surface. When released, what is the maximum speed attained by the block?

Accepted Answer

$v_{max} = \sqrt{\frac{k}{m}} x_0$

Question 4

A box slides across a horizontal surface with friction and compresses a spring. If the coefficient of kinetic friction is $\mu_k$, the box's initial speed is $v_0$, and the spring constant is $k$, what is the maximum compression $x_{max}$ of the spring?

Accepted Answer

$x_{max} = \sqrt{\frac{m v_0^2}{k} - \frac{2\mu_k m g x_{max}}{k}}$

Question 5

A wedge of mass $M$ and a block of mass $m$ are released from rest as shown. The wedge is on a frictionless surface and the block slides down the wedge. What is the speed of the block relative to the ground when it reaches the bottom?

Accepted Answer

$v_b = \sqrt{\frac{2gR M}{M + m}}$

Question 6

A projectile of mass $m_0$ explodes at the top of its trajectory into two fragments, $m_1$ and $m_2$. If $m_1$ lands at the same distance from the launch point as if there were no explosion, how far from the launch point does $m_2$ land?

Accepted Answer

$x_2 = \frac{m_0}{m_2} x_1$

Question 7

A block slides down a frictionless incline of height $h$ and then moves across a rough horizontal surface with coefficient of kinetic friction $\mu_k$. How far does the block travel on the rough surface before coming to rest?

Question 8

A block of mass $m$ is attached to a spring (constant $k$) and slides on a frictionless surface. If the block is released from rest at $x = A$, what is the total mechanical energy of the system?

Accepted Answer

$E = \frac{1}{2} k A^2$

Question 9

A block of mass $m$ slides down a frictionless track and collides with a spring of constant $k$. If the block starts from height $h$, what is the maximum compression $x_{max}$ of the spring?

Accepted Answer

$x_{max} = \sqrt{\frac{2 m g h}{k}}$

Question 10

A block of mass $m$ is projected up a rough incline (angle $	heta$) with initial speed $v_0$. The coefficient of kinetic friction is $\mu_k$. What is the maximum distance $d$ the block travels up the incline before coming to rest?

Accepted Answer

$d = \frac{v_0^2}{2g(\sin	heta + \mu_k \cos	heta)}$

Question 11

A block of mass $m$ slides down a frictionless incline and compresses a spring (constant $k$) at the bottom. If the block starts from rest at height $h$, what is the speed of the block just before it contacts the spring?

Accepted Answer

$v = \sqrt{2gh}$

Question 12

A block of mass $m$ is released from rest at the top of a frictionless ramp of height $h$. At the bottom, it collides elastically with a stationary block of mass $2m$. What is the speed of the first block after the collision?

Accepted Answer

$v = -\frac{1}{3}\sqrt{2gh}$

Question 13

A block of mass $m$ is attached to a spring (constant $k$) and oscillates on a frictionless surface. If the amplitude is $A$, what is the maximum kinetic energy of the block?

Accepted Answer

$K_{max} = \frac{1}{2} k A^2$

Question 14

A block of mass $m$ is projected horizontally with speed $v_0$ onto a rough surface (coefficient of kinetic friction $\mu_k$). What is the distance the block travels before coming to rest?

Accepted Answer

$d = \frac{v_0^2}{2\mu_k g}$

Question 15

A block of mass $m$ is compressed against a spring (constant $k$) by $x_0$ and released on a rough surface (coefficient $\mu_k$). What is the speed of the block after it has moved a distance $d$?

Accepted Answer

$v = \sqrt{\frac{k}{m}(x_0^2 - (x_0 - d)^2) - 2\mu_k g d}$

Question 16

A block of mass $m$ is released from rest at the top of a frictionless ramp of height $h$. At the bottom, it collides inelastically with a stationary block of mass $m$ (they stick together). What is their speed immediately after the collision?

Accepted Answer

$v = \frac{1}{\sqrt{2}}\sqrt{2gh}$

Question 17

A block of mass $m$ is projected up a frictionless incline (angle $	heta$) with initial speed $v_0$. What is the maximum height $h$ reached by the block?

Accepted Answer

$h = \frac{v_0^2}{2g}$

Question 18

A block of mass $m$ is attached to a spring (constant $k$) and oscillates with amplitude $A$. What is the speed of the block when it passes through the equilibrium position?

Accepted Answer

$v = \sqrt{\frac{k}{m}} A$

Question 19

A block of mass $m$ is released from rest at the top of a frictionless ramp of height $h$. At the bottom, it collides elastically with a stationary block of mass $m$. What is the speed of the second block after the collision?

Accepted Answer

$v = \sqrt{2gh}$

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