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?

Accepted Answer

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?

Accepted Answer

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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