**Solved exercises** worksheet

Test what you have learned in the subject *Newton's laws of motion* with this list of exercises with their respective solutions and classified by sections.

### Decomposition of Forces

#### Finding out the angled force with axis

A force is given by the following expression: $\overrightarrow{F}=3\xb7\overrightarrow{i}+2\xb7\overrightarrow{j}$. Calculate angle of horizontal.

#### Force from angle and magnitude

Determine the analytic expression of a force knowing that it forms an 120º angle on X – axis and it has a 5 N magnitude.

### Concurrent and Parallel Forces

#### Concurrent or parallel forces?

Determine if they are concurrent or non concurrent forces (parallel).

### Resultant Force of a System of Forces

#### Massive force

If the following forces act on a body:$${\overrightarrow{F}}_{1}=-10\xb7\overrightarrow{i}+10\xb7\overrightarrow{j}\vdots {\overrightarrow{F}}_{2}=4\xb7\overrightarrow{i}-3\xb7\overrightarrow{j}\vdots {\overrightarrow{F}}_{3}=5\xb7\overrightarrow{i}-\overrightarrow{j}\vdots {\overrightarrow{F}}_{4}=\overrightarrow{i}-\overrightarrow{j}$$

Find out a force whose effect is equal to exert the 4 forces proposed.

#### Adittion of forces on the same direction

Two friends, a stocky one and a slim one, push a coach on the same direction. The first one exert a force of 11 N and the second one a force of 7 N. What is the resultant force?

#### Addition of forces on opposite directions

A boy and a girl tight two cords to a ring and play to find out who is stronger. The boy grab one of the cords and exert a force of 11 N, the girl applies 13 N at the same time. If both pull the cords on different directions... Who will win?

### Linear Momentum

#### Linear Momentum

On a given instant of time, a rocket has a velocity of $\overrightarrow{v}=12\xb7\overrightarrow{i}-7\xb7\overrightarrow{j}m/s\phantom{\rule{0ex}{0ex}}$. Knowing that its mass is 2kg. What is the momentum on that instant?

#### Linear Momentum of several particles

Two pool balls have the following velocity:

Knowing that m_{1} = 170 g y m_{2} = 156 g, calculate the system´s linear momentum formed by both balls.

### Newton´s Second Law

#### Force and trajectory

If the equation of trajectory of a body is $\overrightarrow{r}={t}^{2}\overrightarrow{i}-5t\overrightarrow{j}-2{t}^{2}\xb7\overrightarrow{k}\phantom{\rule{0ex}{0ex}}$ m, What is the force that makes it move if its mass is 4kg?

#### Acceleration from forces

The following forces act on a body:

$${\overrightarrow{F}}_{a}=\overrightarrow{i}+\overrightarrow{j}N;{\overrightarrow{F}}_{b}=\overrightarrow{j}+\overrightarrow{k}N;{\overrightarrow{F}}_{c}=\overrightarrow{i}+\overrightarrow{k}N$$

What is the acceleration of the body if its mass is 2kg?

#### Relationship between masses based on Newton´s second law

Two bodies have acceleration 6 m/s^{2} y 9 m/s^{2} respectively, if they are under the same force. What is the relationship between masses?

#### Traveled distance by a body from force

A 6kg body initially on repose experiences a force of 10 N for 4 seconds. What distance does it travel?

#### Newton´s second law and the nail in the wall

Pablito is nailing a nail into the wall by using a 2kg hammer of mass. The impact speed is 6m/s. What is the resistance force of the wood if the nail is 5mm depth?

### Newton´s Third Law

#### Newton´s third law

Can you say what does a ball on a table interact with and where the forces are exerted at each interaction?

### Impulse

#### Impulse

A 1kg body is subjected to a single force $\overrightarrow{F}=-6\xb7\overrightarrow{i}+8\xb7\overrightarrow{j}N$ for 5s. If we know that its initial velocity is $\overrightarrow{{v}_{o}}=15\xb7\overrightarrow{i}-10\xb7\overrightarrow{j}N$. Calculate:

a)Impulse

b)Linear momentum before and after the force is exerted.

#### Several forces´s impulse

A 35 kg body at rest is subjected to the following forces for 3 minutes:

$${\overrightarrow{F}}_{1}=125\xb7\overrightarrow{i}+200\xb7\overrightarrow{j}\hspace{0.17em};\hspace{0.17em}{\overrightarrow{F}}_{2}=-400\xb7\overrightarrow{i};\hspace{0.17em}{\overrightarrow{F}}_{3}=-150\xb7\overrightarrow{i}+180\xb7\overrightarrow{j}$$

If N is the measure given, calculate:

1. Resultant force

2. Impulse

3. Linear momentum variation

Velocity aquired by the body during these 3 minutes. La velocidad que ha adquirido el cuerpo in these 3 minutes.

#### Impulse from linear momentum variation

A 50 g rubber ball is hit by a racket. Ball´s velocity before the hit is ${\overrightarrow{v}}_{i}=-20\xb7\overrightarrow{i}-20\overrightarrow{j}m/s$,after the hit, the velocity is ${\overrightarrow{v}}_{f}=15\xb7\overrightarrow{i}+20\overrightarrow{j}m/s$ .

Determine the impulse exerted by the racket on the ball and the value of the force, assumed constant, if the are in contact for 0.05 s.

#### Force from impulse

Alice has had a traffic accident when she was driving at 90 km/h. Thanks she was wearing the seatbelt she survived. Can you say what average force has the seatbelt exerted if the impact last 0.05 s and Alice weights 55 kg?

#### Interpretation of impulse in graphs

The following figure represents 3 different forces acting on 3 bodies of equal mass on identical conditions. Knowing that the 3 of them start on rest position, what of them will gain more final speed?

### Momentum Conservation Principle

#### Conservation of linear momentum principle

A competitor shoots a plate by using a rifle of 2,5 Kg on a skeet shooting competition. Knowing that the ball is 23 g and it is shot horizontally at 350 m/s, what is the recoil of the rifle?

#### Collision velocity and momentum conservation

A 3 Kg ball is moving towards left at 6 m/s. It collides with a 5 Kg ball moving towards right at 2 m/s. After the collision the second ball went through the left at 3 m/s. Calculate the velocity of the first ball after the crash.

#### Momentum conservation in explosions

A 10 Kg body is moving at ${\overrightarrow{v}}_{i}=\overrightarrow{i}+\overrightarrow{j}m/s$ and after its explosion it breaks into 3 pieces. The first piece is 3 Kg and goes through to ${\overrightarrow{v}}_{1}=\frac{1}{2}\xb7\overrightarrow{i}+2\xb7\overrightarrow{j}m/s$, the second body weights 5 Kg and goes through to ${\overrightarrow{v}}_{2}=2\xb7\overrightarrow{i}+\frac{1}{2}\xb7\overrightarrow{j}m/s$. What velocity does the third piece reach?

#### Momentum conservation - variable mass

A truck moves 40 Km/hr in a straight line over a conventional road. An open container is placed on the truck. The total weight is 1800 Kg. When it starts to rain, the container got filled with water at 6 L/ min. Paying no heed to friction , what is the speed of the truck after one hour of rain?