Physical quantities, units and vectors

A regional train travels at $72km/h$ and a runner at $15m/s$. Express the speed of the train in $m/s$ and that of the runner in $km/h$.

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Seawater has a density of $1.03g/c{\mathrm{m}}^3$. Express the density in $kg/{\mathrm{m}}^3$ and compute the mass of $2.5\mathrm{L}$ of seawater.

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Express in SI units, in scientific notation, the volume $V=350\mathrm{c}{\mathrm{m}}^3$, the area $A=2.4\mathrm{m}{\mathrm{m}}^2$ and the concentration $72g/L$.

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A rectangular sheet measures $21.0\mathrm{cm}$ by $29.7\mathrm{cm}$. Compute its area in ${\mathrm{m}}^2$, in scientific notation and with the correct number of significant figures.

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Treating the Earth as a sphere of radius $R=6.37\cdot 10^6\mathrm{m}$, compute its volume and state its order of magnitude.

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The air resistance on a fast body is written $F=\beta v^2$, where $F$ is a force and $v$ a velocity. Determine the dimensions and the SI unit of the coefficient $\beta$.

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The speed $v$ with which an object dropped from rest reaches the ground depends on the gravitational acceleration $g$ and on the height $h$. Assuming $v=k{g}^a{h}^b$ with $k$ dimensionless, determine $a$ and $b$.

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A force of magnitude $25\mathrm{N}$ forms an angle of $210^{\circ }$ with the positive $x$-semiaxis, measured counterclockwise. Compute its Cartesian components.

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A velocity has components $v_x=-6.0m/s$ and $v_y=8.0m/s$. Determine its magnitude and the angle with the positive $x$-semiaxis, measured counterclockwise.

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Three forces in the $xy$-plane act on a point particle: $40\mathrm{N}$ at $0^{\circ }$, $30\mathrm{N}$ at $90^{\circ }$ and $20\mathrm{N}$ at $225^{\circ }$, with the angles measured counterclockwise from the positive $x$-semiaxis. Determine the magnitude and direction of the resultant.

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A body undergoes the displacement $\vec{s}=(3.0\hat{ı}+4.0\hat{ȷ})\mathrm{m}$ under the action of the constant force $\vec{F}=(5.0\hat{ı}-2.0\hat{ȷ})\mathrm{N}$. Compute the work of the force and the angle between the force and the displacement.

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The force $\vec{F}=(10\hat{ı}-5.0\hat{ȷ}+20\hat{k})\mathrm{N}$ is applied at the point with position $\vec{r}=(0.30\hat{ı}+0.40\hat{ȷ})\mathrm{m}$ relative to the pole $O$. Compute the torque of the force about $O$ and its magnitude.

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The position of a point is $\vec{r}(t)=1.5t^2\hat{ı}+\left(2.0t^3-6.0t\right)\hat{ȷ}$, with lengths in meters and time in seconds. Compute the velocity and the acceleration at the instant $t=2.0\mathrm{s}$ and their magnitudes.

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