This problem requires us to determine the** acceleration **and **final velocity** after some displacement in an inclined plane.

Remember that for objects on inclined planes, we use a coordinate system where __the + x-axis points parallel to the ramp__. The object accelerates due to a

$\overline{){\mathbf{a}}{\mathbf{=}}{\mathbf{g}}{\mathbf{}}{\mathbf{s}}{\mathbf{i}}{\mathbf{n}}{\mathbf{}}{\mathbf{\theta}}}$

where *θ* is the angle between the plane's surface and the horizontal. *This acceleration is positive*, since the component of gravity is acting along the positive *x*-axis.

Since the acceleration is constant, we can apply the four kinematics equations. As always for kinematics problems, we'll follow these steps:

**Identify**the,__target variable__, and Unknowns for each part of the problem—remember that__knowns__*only*(Δ**3**of the**5**variables*y*,*v*_{0},*v*,_{f}*a*, and*t*)*are needed*to solve any kinematics problem, and we already know what*a*is.__Choose a UAM equation__with**only one unknown**, which should be our**target variable**.__Solve the equation__for the target variable, then__substitute known values__and__calculate__the answer.

The four UAM equations are:

$\overline{)\mathbf{}{{\mathbf{v}}}_{{\mathbf{f}}}{\mathbf{}}{\mathbf{=}}{\mathbf{}}{{\mathbf{v}}}_{{\mathbf{0}}}{\mathbf{}}{\mathbf{+}}{\mathbf{a}}{\mathbf{t}}\phantom{\rule{0ex}{0ex}}{\mathbf{\u2206}}{\mathit{x}}{\mathbf{=}}{\mathbf{}}\mathbf{\left(}\frac{{\mathbf{v}}_{\mathbf{f}}\mathbf{+}{\mathbf{v}}_{\mathbf{0}}}{\mathbf{2}}\mathbf{\right)}{\mathbf{t}}\phantom{\rule{0ex}{0ex}}{\mathbf{\u2206}}{\mathit{x}}{\mathbf{=}}{\mathbf{}}{{\mathbf{v}}}_{{\mathbf{0}}}{\mathbf{t}}{\mathbf{+}}{\frac{1}{2}}{\mathbf{a}}{{\mathbf{t}}}^{{\mathbf{2}}}\phantom{\rule{0ex}{0ex}}{\mathbf{}}{{{\mathbf{v}}}_{{\mathbf{f}}}}^{{\mathbf{2}}}{\mathbf{=}}{\mathbf{}}{{{\mathbf{v}}}_{{\mathbf{0}}}}^{{\mathbf{2}}}{\mathbf{}}{\mathbf{+}}{\mathbf{2}}{\mathbf{a}}{\mathbf{\u2206}}{\mathit{x}}}$

**(a)** First, the problem asks for the acceleration of the block. We'll use the equation *a* = *g* sin*θ*. Assuming *g* = 9.81 m/s^{2}:

$\mathbf{a}\mathbf{=}\mathbf{(}\mathbf{9}\mathbf{.}\mathbf{81}\mathbf{}\raisebox{1ex}{$\mathbf{m}$}\!\left/ \!\raisebox{-1ex}{${\mathbf{s}}^{\mathbf{2}}$}\right.\mathbf{)}\mathbf{}\mathbf{s}\mathbf{i}\mathbf{n}\mathbf{(}\mathbf{23}\mathbf{.}\mathbf{0}\mathbf{\xb0}\mathbf{)}$

The block shown in the figure has mass *m* = 7.5 kg and lies on a fixed smooth frictionless plane tilted at an angle *θ* = 23.0° to the horizontal.**(a)** Determine the acceleration of the block as it slides down the plane.**(b)** If the block starts from rest 13.0 m up the plane from its base, what will be the block's speed when it reaches the bottom of the incline?

Frequently Asked Questions

What scientific concept do you need to know in order to solve this problem?

Our tutors have indicated that to solve this problem you will need to apply the Inclined Planes concept. You can view video lessons to learn Inclined Planes. Or if you need more Inclined Planes practice, you can also practice Inclined Planes practice problems.