Question

(a) What is the magnitude of the momentum of a 10,000-kg truck whose speed is 12.0 m/s? (b) What speed would a 2000-kg SUV have to attain in order to have (i) the same momentum?(ii) the same kinetic energy?

Short answer

(a) The magnitude of the momentum of the truck is $12\times {10}^4\mathrm{kgm}/\mathrm{s}$.

(b) (i) To have the same momentum as the truck speed of the SUV will be

60 m/s.

(ii)To have the same kinetic energy as the truck speed of the SUV will be 26.38 m/s.

Step-by-step solution

Given information in the question.

Mass of the truck, m =10,000kg

Speed of the truck, v =12m/s

Mass of the SUV,$m_2=2,000\mathrm{kg}$

Definition of kinetic energy

Kinetic energy, which may be seen in the movement of an object, particle, or group of particles, is the energy of motion.

Finding the magnitude of the momentum of the truck for part (a)

As we know the momentum of an object is

$\overrightarrow{p}=m\overrightarrow{v}$

Substituting the values into the formula

$$\begin{gathered} \overrightarrow{p}=m\overrightarrow{v} \\ =10000kg\times 12m/s \\ =12\times {10}^{4}kg.m/s \end{gathered}$$

Hence the magnitude of the momentum of the truck is $12\times {10}^4\mathrm{kg}.\mathrm{m}/\mathrm{s}$.

Finding the speed of the SUV for the same momentum for part (b)

$$\begin{gathered} v_2=? \\ {p}_1=12\times {10}^4\mathrm{kg}.\mathrm{m}/\mathrm{s} \end{gathered}$$

According to the question, momentum is the same

$$\begin{gathered} p_1=p_1 \\ {p}_1=m_2{v}_2 \\ 12\times {10}^{4}kg.m/s=2,000kg\times v_2 \\ {v}_2=60m/s \end{gathered}$$

Therefore, toattainthe same momentum as the truck speed of the SUV will be 60m/s.

Finding the speed of the SUV for the same kinetic energy for part (b)

$m_1=10,000\mathrm{kg}$,

$$\begin{gathered} v_1=12\mathrm{m}/\mathrm{s} \\ {m}_2=2,000\mathrm{kg} \\ {v}_2=? \end{gathered}$$

According to the question, kinetic energy is the same

$$\begin{gathered} {\mathrm{K}}_1={\mathrm{K}}_2 \\ \frac{1}{2}{\mathrm{m}}_1{\mathrm{v}}_1^2=\frac{1}{2}{\mathrm{m}}_2{\mathrm{v}}_2^2 \\ \frac{1}{2}\left(10000\mathrm{kg}\right){\left(12\mathrm{m}/\mathrm{s}\right)}^2=\frac{1}{2}\left(2000\mathrm{kg}\right){\mathrm{v}}_2^2 \\ {\mathrm{v}}_2^2=\sqrt{\frac{\left(10000\mathrm{m}/\mathrm{s}\right){\left(12\mathrm{kg}\right)}^2}{2000\mathrm{kg}}} \\ {\mathrm{v}}_2=26.38\mathrm{m}/\mathrm{s} \end{gathered}$$

Therefore, to attain the same kinetic energy as the truck speed of the SUV will be 26.38m/s .