Two forces, one 30 N, the other 20 N, act on a body. The directions of the forces are unknown.
What is the minimum possible magnitude of the resultant vector?

The minimal possible value occurs if the vectors are in opposite directions.

30 - 20 = 10 N

Two forces, one 20 N, the other 15 N, act on a body. The directions of the forces are unknown.
What is the maximum possible magnitude of the resultant vector?

The maximum possible value occurs if the vectors are in the same direction.

20 + 15 = 35 N.

A man walks 3 km south and 4 km west.

What is the magnitude of his displacement from the original position?

Input the right answer in kilometers.

$$R= {\sqrt{a^2 +b^2} } = \sqrt{9 +16} = 5$$

A man walks 5 km north and 12 km east.

What is the direction from the north of his displacement from his start position?

Give the answer in degrees to 1 decimal point

$$Tan(\theta) = {opposite\over adjacent} = 12/5$$

$$\theta = ArcTan(12/5) = {67.4 }$$ degrees.

Two forces, F(a) and F(b), act on the object. F(a) is a force of 7 N acting vertically upwards, F(b) acting on the object 24 N due east.

What is the magnitude of the resultant force F?
Give the answer in newtons to 1 decimal point.

Vector F(a) added to vector F(b) gives the resultant vector F.

$$F = {\sqrt{F(a)^2 +F(b)^2}} = {\sqrt{7^2 + 24^2}} = 25$$

Two forces, F(a) and F(b), act on the object. F(a) is a force of 5 N acting vertically downwards, F(b) acting on the object 7 N due west.

What is the magnitude of the resultant force F?
Give the answer in newtons to 1 decimal place.

Vector F(a) added to vector F(b) gives the resultant vector F.

$$F = {\sqrt{F(a)^2 +F(b)^2}} = {\sqrt{5^2 + 7^2}} \approx 8.6$$

A swimmer sets off to cross straight the river 20 m wide. If he travels at speed of 0.5 m/s and the speed of river flow is 2 m/s, how far down the river from his starting position will he land?

Give the answer in meters to 1 decimal place.

Time taken to cross the river: $$t = 20/2 = 10 s$$

Distance travelled by the swimmer down the river: $$s = 10*2 = 20 m$$

A motor boat can travel in still water with a speed of 10 m/s.Calculate its speed relative to a fixed observer on the shore in the case when the boat sails downstream the river with current of 2 m/s.

Give the answer in m/s to 1 decimal place.

As the boat sails downstream, the vector of its velocity and the vector of current are codirectional, therefore, observed velocity = velocity of the boat + current = 10 + 2 = 12 m/s

Scalar and vector quantities 1.1

In which set are all the quantities scalar?

Vector quantities in the question are shown in bold:

1. Time, displacement, velocity
2. Acceleration, velocity, kinetic energy
3. Mass, gravitational potential energy, work
4. Volume, work, weight
Displacement, velocity and acceleration - 1.1

The graph below shows the motion of the cyclist as he accelerates from rest. What is the acceleration?

$$a= ∆v⁄∆t=15/30=0.5 m /s^2$$

Displacement, velocity and acceleration - 1.2

The graph below shows the motion of the cyclist as he accelerates from rest. What is the distance travelled by him during the first 30 s?

Input the distance in meters

In order to find the distance, the area under motion graph should be calculated. In this particular example, it is the area of the triangle shown in red in the plot below:

$$s = (∆v*∆t)/2=(30*15)/2=225$$

Displacement, velocity and acceleration - 1.3

The plot below shows the distance travelled by the jet plane in one minute of its flight. What is the plane’s velocity?

Input the correct answer in m/s to one decimal place.

The velocity is $$v = {\Delta s\over{\Delta t}}=30000/60=500\space m/s$$

Equations for uniformly accelerated motion in one dimension - 1

Choose the correct evaluation of the formula $$v = u + at$$ in order to calculate time, t.

$$t = \frac{v -u}{a}$$

Equations for uniformly accelerated motion in one dimension - 2

Choose the correct evaluation of the formula $$v = u + at$$ in order to calculate acceleration, a.

a =  \frac{u-v }{t}

Equations for uniformly accelerated motion in one dimension - 3

Choose the correct evaluation of the formula $$s = ut + {at^2\over 2}$$ in order to calculate acceleration, a.

$$a = { 2(s - ut)\over t^2}$$

Equations for uniformly accelerated motion in one dimension - 4

Choose the correct evaluation of the formula $$s = ut + {at^2\over 2}$$ in order to calculate initial velocity, u.

$$u = s/t -at/2$$

Equations for uniformly accelerated motion in one dimension - 6

Choose the correct evaluation of the formula $$v^2 = u^2+ 2as$$ in order to calculate acceleration, a.

$$a ={ {v^2 - u^2}\over2s}$$

Equations for uniformly accelerated motion in one dimension - 7

Choose the correct evaluation of the formula $$v^2 = u^2+ 2as$$ in order to calculate displacement, s.

$$s = {{v^2 - u^2}\over 2a}$$

Uniformly accelerated motion - 1

A hockey puck has crossed a 60-meter long arena in 3 seconds and stopped. Calculate the initial velocity that was imparted to the puck.

Give the answer in m/s to 1 decimal place.

Equations for uniformly accelerated motion in one dimension:

$$v^2 = u^2+ 2as {(1)}$$ and $$v = u + at(2)$$.

The puck had stopped, so $$v = 0$$. In this case, $$(2) \Leftrightarrow a = -u/t$$, therefore, $$(1) \Leftrightarrow u^2 - \frac{2u}{t}s=0 \Leftrightarrow u(u-2s/t)=0$$. This equation has two roots, the first one, u = 0, doesn't agree with the statement of the problem and should be rejected; the other one, $$u = 2s/t$$, is correct.

$$u = 2s/t = 2*60/3 = 40$$

Uniformly accelerated motion - 2.

A train approaches the station at 5 m/s, and it takes 1 minute to stop. With what acceleration was it braking?

Give the answer in m/s to 3 decimal places.

$$v = u + at$$$$v = 0,$$therefore $$a = -u/t = 5/60 \approx 0.083$$

Uniformly accelerated motion - 3.

A train accelerates at $$0.9 m s^{-2}$$ for 10 seconds. What is the final velocity?

Give the answer in m/s to 1 decimal place.

$$v = u + at, u =0,$$therefore, $$v = 0.9*10 = 9$$.

Defenitions

The definition of mass is

Mass is the quantity of inertia possessed by an object.

Definitions - 2

Select the correct definition of acceleration.

Acceleration is the rate of change of velocity as a function of time.

Definitions - 3

The definition of force is...

Force is the quantitative description of the interaction between two physical bodies.

Definitions - 4

Select the correct definition of velocity.

Velocity is a vector measurement of the rate and direction of motion.

Definitions - 5

The definition of mechanical work is...

Work is the integral of the force over a distance of displacement.

Definitions - 6

The definition of mechanical power is...

Power is the time rate at which work is done.

Definitions - 7

The definition of energy is...

Energy is the capacity of a physical system to perform work.

Non-uniform Acceleration

The motion graph of a lorry is shown below. Calculate the displacement the lorry would travel in 8 seconds.

Input the correct answer in meters to 1 decimal place.

To calculate the displacement, the area under velocity-time graph should be found. In this case, it is the sum of the areas of two triangles and one rectactangle: $$S = {(4*4)\over2} + 2*4+ {(2*4)\over2} = 8+8 +4 = 20$$

Free-body diagram

A car goes at constant speed along the straight road. Select the correct free-body diagram for it.

The driving force is directed towards the course of a car, the drag force counteracts the movement and is directed backwards. The gravitational force pulls the car down, and normal reaction force that counteracts gravitational force is directed upwards.

Centre of gravity

Select the correct definition for centre of gravity.

Centre of gravity is the point through which all the weight of an object appears to act. And centre of mass is the point through which all the mass of an object appears to be concentrated.

Centre of mass

Select the correct definition for centre of mass.

Centre of mass is the point through which all the mass of an object appears to be concentrated. Centre of gravity is the point through which all the weight of an object appears to act.

Newton's first law of motion

Select the correct statement of Newton's first law of motion.

Newton's First Law of MotionAn object remains at rest or continues moving along a straight line at constant velocity unless acted on by an external force.

Newton's Second Law of Motion - The acceleration of an object is directly proportional to the magnitude of the net force on the object in the same direction as the net force, and inversely proportional to the mass of the object.

Newton's Third Law of Motion - If body A exerts a force on body B, then body В exerts a force of the same size on body A, but in the opposite direction.

Newton's second law of motion

Select the correct statement of Newton's second law of motion.

Newton's First Law of Motion - An object remains at rest or continues moving along a straight line at constant velocity unless acted on by an external force.

Newton's Second Law of MotionThe acceleration of an object is directly proportional to the magnitude of the net force on the object in the same direction as the net force, and inversely proportional to the mass of the object.

Newton's Third Law of Motion - If body A exerts a force on body B, then body В exerts a force of the same size on body A, but in the opposite direction.

Newton's third law of motion

Select the correct statement of Newton's third law of motion.

Newton's First Law of Motion - An object remains at rest or continues moving along a straight line at constant velocity unless acted on by an external force.

Newton's Second Law of Motion - The acceleration of an object is directly proportional to the magnitude of the net force on the object in the same direction as the net force, and inversely proportional to the mass of the object.

Newton's Third Law of Motion - If body A exerts a force on body B, then body В exerts a force of the same size on body A, but in the opposite direction.

Gravitational field strength

An astronaut on Mars has a weight of 297.6N and a mass of 80kg. What is the gravitational field strength on Mars?

Input the correct answer in $$m s^{-2}$$ to 2 decimal points.

Gravitational field strength, $$g = W/m =297.6/80 =3.72 ms^{-2}$$

Weight

Calculate a weight on the Moon of a man with a mass of 80kg, g on the Moon is $$1.63 ms^{-2}$$.

$$W = mg_{Moon} = 130.4N$$

Pair of forces

A ball with a mass of 1 kg elastically collides with another ball of unknown mass. After collision, balls have obtained accelerations of $$1 ms^{-2}$$ and $$2 ms^{-2}$$, respectively. Calculate the mass of the second ball.

Input the answer kg to 1 decimal point.

By Newton's third law of motion, during the collision the forces, by which the balls interact with each other, are the same in size, but opposite in direction: $$F_1 = -F_2$$. Using the fact that the forces are equal in size, $$m_1a_1 = m_2a_2\Rightarrow m_2=\frac{m_1a_1}{a_2}=1*1/2=0.5$$

Kinetic energy

Calculate the kinetic energy of a bullet with a mass of 9g and a speed of 900 m/s.

Input the correct answer in J to 1 decimal point.

$$E_k = {mv^2\over 2}= 9*10^{-3}*81*10^4/2 = 7290J$$

Gravitational potentional energy

A barrel with a mass of 100kg was lifted by crane to a height of 20m. Calculate the increase of gravitational potential energy.

Input the correct answer in J to one decimal point. g = 9.81 $$m/s^2$$

$$\Delta E_{grav}=mg\Delta h = 100*9.81*20 =19620J$$

Conservation of energy

A ball with a weight of 100g was dropped from a height of 100m. Calculate the speed of the ball at a height of 20m. Ignore air resistance effects when answering.

Input the correct answer in m/s to one decimal place.

$$\Delta E_k+\Delta E_{grav} = 0 \Rightarrow mgh_0 = mgh_1 + {mv^2 \over 2}\Rightarrow v=\sqrt{2g(h_0 -h_1)}\approx19.8 \space (m/s)$$

Work

A boy pulls a wagon with a handle that is at an angle of $$30^{\circ}$$ with the ground. If he pulls it with a force of 300N, how much work will he do over the distance of 100 m?

Input the answer in kJ to one decimal place.

The mechanical work is $${A = FS\cos{\alpha}}$$, where $$\alpha$$is the angle between the force applied to the object and displacement. $$A = 300*100*\cos{30^{\circ}}\approx26.0 \space kJ$$

Law of conservation of energy

Select the correct statement of the law of conservation of energy.

Law of conservation of energy states that energy cannot be created or destroyed.

Scalar and vector quantities 1.1-2

In which set are all the quantities scalar?