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These problems about momentum are designed to enhance your understanding of momentum and its definition, as well as to provide practical applications of its formula in various situations.

Momentum is defined as the product of an object's mass and its velocity, represented by the formula \[\vec{P}=m\vec{v}\] Like velocity, momentum is a vector quantity in the same direction as the object's velocity.

The SI units of momentum are kilogram-meters per second ${\rm kg.m/s}$.

An object can possess a large momentum if it has either a large mass or a high velocity.

When solving momentum problems, the first step is to assign a positive direction. Then, compare the velocities in this direction. Velocities in the same direction as the positive direction are considered to have positive momentum, while those in the opposite direction have negative momentum.

*These problems are straightforward and beneficial for students in classes 10 and 11. For more complex problems appropriate for class 12 or college level, please refer to the sections on momentum and impulse problems.*

Momentum Problems

**Problem (1): What is the definition of momentum in physics? **

**Solution:** the product of a particle's mass and velocity in physics is called the particle's momentum, $\vec{p}=m\vec{v}$. Momentum is a vector quantity like velocity, acceleration, and force. The units of momentum are kg.m/s.

**Practice Problem (2): If a truck has a mass of 1000 kilograms and is traveling with a speed of v = 50 m/s, what is its momentum?**

**Solution:** By definition, momentum is the product of mass and velocity, represented by the formula $P=m\,v$. Thus, substituting the given numerical values into this, we have \begin{align*} P&=mv\\&=1000\times 50\\&=50,000\quad {\rm kg.m/s}\end{align*} In all momentum problems, the question typically asks the magnitude of the momentum, regardless of its direction.

**Problem (3): If a car has a mass of 2000 kilograms and travels with a velocity of v = 72 km/h, what is its momentum?**

**Solution:** Momentum is mass times velocity, $p=mv$ and its units in the SI system are $\rm kg.m/s$. In this problem, we first convert the units of velocity to SI units (i.e., from $\rm km/h$ to $\rm m/s$ by multiplying it by $\frac {10}{36}$). Therefore, the momentum of the car is calculated as follows: \begin{align*} P&=mv\\&=2000\times \Big(72\times \frac{10}{36}\Big)\\&=40,000\quad {\rm kg.m/s}\end{align*} This means the car has a momentum of 40,000 kg.m/s.

**Problem (4): An 8-kilogram bowling ball is rolling in a straight line toward you. If its momentum is 16 kg·m/sec, how fast is it traveling?**

**Solution:** using momentum formula $p=mv$ and solving for the velocity, we have \begin{align*} v&=\frac pm\\&=\frac{16}{8}\\&=2\quad {\rm m/s} \end{align*}

**Problem (5): A toy dart gun generates a dart with a momentum of 140 kg.m/s and a velocity of 4 m/s. What is the mass of the dart in grams? **

**Solution:** by applying momentum formula $p=mv$ and solving for $m$, we get \begin{align*} m&=\frac pv\\&=\frac {140}{4}\\&=35\quad {\rm kg} \end{align*} to convert it to the grams, multiply it by 1000 so the dart's mass is $35,000\,{\rm g}$.

**Problem (6): A beach ball is rolling in a straight line toward you at a speed of 0.5 m/sec. Its momentum is 0.25 kg·m/sec. What is the mass of the beach ball?**

**Solution:** Given that the momentum is $0.25\,\rm kg\cdot m/s$ and the velocity is $0.5\,\rm m/s$, we can rearrange the formula $p=mv$ to solve for the mass of the ball: \begin{align*} m&=\frac pv\\&=\frac {0.25}{0.5}\\&=0.5\quad {\rm kg} \end{align*}

**Problem (7): A 5,000-kilogram truck travels in a straight line with a speed of 54 km/h. What is its momentum?**

**Solution:** The SI units of momentum are $\rm kg\cdot m/s$. Therefore, we first convert the speed units into SI units by multiplying them by $\frac {10}{36}$. As a result, the truck's speed becomes $v=54\times \frac {10}{36}=15\,{\rm m/s}$. Subsequently, we calculate the momentum as follows: \begin{align*} p&=mv\\&=5000\times 15\\&=75,000\quad {\rm kg.m/s}\end{align*}

**Problem (8): A 1,400-kilogram car is traveling in a straight line. Its momentum is equal to that of the truck in the previous question. What is the velocity of the car?**

**Solution**: the car's velocity is \begin{align*} v&=\frac pm\\&=\frac{75,000}{1400}\\&=53.6\quad {\rm m/s}\end{align*}

**Problem (9): A bus traveling at a speed of 50 km/h has a momentum of 180,345 kg.m/s. What is the mass of the bus?**

**Solution**: first convert the speed's units to the SI units of velocity ($\rm \frac ms$) by multiplying it by $\frac {10}{36}$. Next, using the formula of momentum $p=mv$ and solving for the mass, we get \begin{align*} m&=\frac pv\\&=\frac{180,345}{50\times \frac{10}{36}}\\&=12984.84\quad {\rm kg}\end{align*}

**Problem (10): Which would take more force to stop in 10 seconds: an 8.0-kilogram ball rolling in a straight line at a speed of 0.2 m/sec or a 4.0-kilogram ball rolling along the same path at a speed of 1.0 m/sec?**

**Solution**: According to Newton's second law of motion, the change in an object's momentum is directly proportional to the force applied. As a result, an object with higher momentum requires a greater force to bring it to a stop.

Consider a $8\,{\rm kg}$-ball with a momentum calculated as $p=mv=(8)(0.2)=1.6\,{\rm kg.m/s}$. On the other hand, a $4\,{\rm kg}$-ball has a momentum of $p=mv=(4)(1)=4\,{\rm kg.m/s}$. Therefore, a larger force is required to stop the 4 kg ball due to its higher momentum.

**Problem (11): The momentum of a truck traveling in a straight line at 15 m/sec is 41,500 kg·m/sec. What is the mass of the truck?**

**Solution**: using the momentum formula, we have \begin{align*} m&=\frac pv\\&=\frac {41,500}{15}\\&=2766.6\quad {\rm kg}\end{align*}

**Problem (12): The parking brake on a 1500 kg automobile has broken, and the car has reached a momentum of 7500 kg.m/s. What is the velocity of the vehicle?**

**Solution**: using the definition of momentum and solving for the velocity, we have \begin{align*} v&=\frac pm\\&=\frac{7500}{1500}\\&=5\quad{\rm m/s}\end{align*}

**Problem (13): A proton with mass of $1.67\times 10^{-27}\,{\rm kg}$ moving at the speed of $5\times 10^{5}\,{\rm m/s}$. What is its momentum?**

**Solution**: momentum is mass multiply by velocity so \begin{align*} p=&mv\\&=(1.67\times 10^{-27})(5\times 10^{5})\\&=8.35\times 10^{-23}\quad {\rm kg.m/s}\end{align*}

**Problem (14): A 0.14-kilogram baseball is thrown in a straight line at a velocity of 30 m/sec. What is the momentum of baseball?**

**Solution**: baseball's momentum, $p=mv$, is determined as \begin{align*} p&=mv\\&=(0.14)(30)\\&=4.2\quad {\rm kg.m/s}\end{align*}

**Problem (15): A pitcher can throw a 0.14-kg baseball with the same momentum as a 3-g bullet moving at a speed of 3000 m/s. What is baseball's speed? **

**Solution**: Since the momentum of the bullet, $p_2=m_2v_2$, is the same as the baseball's one $p_1=m_1v_1$, so equating those and solving for the unknown speed of the baseball, we get \begin{align*} p_1&=p_2\\m_1v_1&=m_2v_2\\\Rightarrow v_2&=\frac{m_1 v_1}{m_2}\\&=\frac{0.003\times 3000}{0.14}\\&=64.28\quad {\rm m/s}\end{align*} in above we converter the bullet's mass to the SI unit of mass $kg$ by dividing it by $1000$.

**Problem (16): Another pitcher throws the same baseball in a straight line. Its momentum is 2.1 kg·m/sec. What is the velocity of the ball?**

**Solution**: using momentum formula, $p=mv$ and solving for the velocity, we have \begin{align*} v&=\frac pm\\&=\frac {2.1}{0.14}\\&=15\quad{\rm m/s} \end{align*}

**Problem (17): A 1-kilogram turtle crawls in a straight line at a speed of 0.01 m/sec. What is the turtle’s momentum?**

**Solution**: momentum is defined as the product of mass and velocity, so \begin{align*} p&=mv\\&=(1)(0.01)\\&=0.01\quad {\rm kg.m/s} \end{align*}

**Practice Problem (18): A bicycle and its rider have a mass of 100 kg. At what speed do they have the same momentum as an 1800-kg car traveling at 2 m/s?**

**Solution**: the momenta of bicycle and car are equal so we have \begin{align*} p_c&=p_b\\m_b\,v_b &= m_c\, v_c \\(100)v_b &= (1800)(10) \\ \Rightarrow v_b &= \frac{1800\times 2}{100}\\&=36\quad {\rm m/s}\end{align*}

To solve introductory momentum problems and questions, you must first learn the definition of momentum as the product of mass and velocity and then apply it to find the unknown. All the above problems are easy and could be solved without any difficulty.

In this article, we solved some simple problems on momentum that are helpful for high school students.

We learned that momentum is defined as the product of an object's mass and its velocity. \[\vec{p}=m\vec{v}\]

Momentum is a vector quantity in physics, like velocity and displacement. It has both a magnitude and a direction.

**Date Created:** 10/26/2020

**Author**: Dr. Ali Nemati

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