Question

A block slides up a 30˚ incline. If µk = 0.4 and the block’s initial speed up the incline is 2 m/s, how far up the incline will the block move before coming to rest?

Answer 1

Answer 2

Given:

• Incline angle ($\theta$) = $30^{\circ }$

• Coefficient of kinetic friction (${\mu }_k$) = 0.4

• Initial speed ($v_0$) = 2 m/s (up the incline)

• Final speed ($v$) = 0 m/s (comes to rest)

• Acceleration due to gravity ($g$) = 9.8 m/s${}^2$

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Step 1: Determine the Acceleration ($a$)

The block experiences two forces slowing it down:

1. Component of gravity along the incline:

   $$F_{\text{gravity}}=mg\sin \theta$$

2. Kinetic friction force:

   $$F_{\text{friction}}={\mu }_k\cdot N={\mu }_k\cdot mg\cos \theta$$

   where $N=mg\cos \theta$ is the normal force.

The net deceleration ($a$) is given by:

$$a=-(g\sin \theta +{\mu }_{k}g\cos \theta )$$$$a=-g(\sin 30^{\circ }+{\mu }_k\cos 30^{\circ })$$$$a=-9.8(0.5+0.4\cdot 0.866)$$$$a=-9.8(0.5+0.3464)$$$$a=-9.8\cdot 0.8464\approx -8.29{\text{m/s}}^2$$

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Step 2: Use Kinematics to Find Distance ($d$)

We use the kinematic equation:

$$v^2=v_0^2+2ad$$

Given $v=0$ and $v_0=2\text{m/s}$:

$$0=(2{)}^2+2(-8.29)d$$$$0=4-16.58d$$$$16.58d=4$$$$d=\frac{4}{16.58}\approx 0.24\text{m}$$

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Answer:

The block will move $\boxed{{\displaystyle 0.24\text{m}}}$ up the incline before coming to rest