The Fundamental Principle of Dynamics

Calculate the net force required for a toy cart of $2\ \text{kg}$ to accelerate at $3\ \text{m/s}^2$.

A net force of $20\ \text{N}$ acts on a box of $5\ \text{kg}$. What acceleration does the box acquire?

A scooter together with its rider accelerates at $2\ \text{m/s}^2$ when a net force of $100\ \text{N}$ acts on the combination. Calculate the total mass.

Using Newton's second law, show that $1\ \text{N}$ is equivalent to $1\ \text{kg} \times \text{m/s}^2$ and give a simple numerical example to illustrate it.

The same force of $50\ \text{N}$ is applied first to a ball of $0.5\ \text{kg}$ and then to a medicine ball of $5\ \text{kg}$, as shown in the figure.

Calculate the acceleration of each one and explain which accelerates more and why.

Two forces of $10\ \text{N}$ and $14\ \text{N}$ push a box of $4\ \text{kg}$ in the same direction, as shown in the figure.

Calculate the net force on the box and its acceleration.

A person pushes a toolbox of $8\ \text{kg}$ with a force of $60\ \text{N}$, and the friction with the ground is $20\ \text{N}$.

1. a) Calculate the net force on the box.
2. b) Calculate its acceleration.

Three forces act on a crate of $10\ \text{kg}$ along the same line: $50\ \text{N}$ to the right, $30\ \text{N}$ to the right and $20\ \text{N}$ to the left.

1. a) Calculate the net force.
2. b) Calculate the acceleration of the crate.
3. c) What additional force would need to be applied so that the crate moves at constant velocity? Reason your answer.

A worker pushes a trolley of $20\ \text{kg}$ with a force of $100\ \text{N}$ and the friction is $20\ \text{N}$.

1. a) Calculate the acceleration of the cart.
2. b) If the worker doubles their push to $200\ \text{N}$ and the friction does not change, does the acceleration double as well? Calculate it and explain why this happens.

A cyclist moving forwards applies the brakes. In which direction does the net force on the bicycle point while braking: forwards or backwards? What about its acceleration? Explain the relationship between the direction of the net force and that of the acceleration.

A car of $1000\ \text{kg}$ brakes with a net force of $5000\ \text{N}$, as shown in the figure.

1. a) Calculate its acceleration and interpret the sign of the result.
2. b) If it was travelling at $20\ \text{m/s}$, calculate how long it takes to come to a stop (use $v = v_0 + a \times t$).

A car of $800\ \text{kg}$ travels at $20\ \text{m/s}$ and needs to stop in $5\ \text{s}$.

1. a) Calculate the braking acceleration.
2. b) Calculate the required net braking force.

An aeroplane flies in a straight line at a constant velocity of $250\ \text{m/s}$. What is the net force on the aeroplane? What is its acceleration? Relate this to Newton's second law of motion.

A sledge slides along horizontal ice and we stop pushing it. A classmate claims: "since there is no longer any force pushing it, it will stop straight away".

1. a) According to Newton's second law, how would the sledge move if friction were exactly zero? What would the net force on it be then?
2. b) On real ice the sledge eventually comes to a stop: which force is responsible and in which direction does it act?
3. c) Explain why the classmate's statement confuses "there is no applied force" with "there is no net force".

A pushchair of $6\ \text{kg}$ and a shopping trolley of $12\ \text{kg}$ experience the same net force of $12\ \text{N}$.

1. a) Calculate the acceleration of each one.
2. b) Compare the masses and accelerations: what relationship do you find between them?

In a laboratory, the acceleration of a trolley is measured when different forces are applied to it: with $2\ \text{N}$ it accelerates at $0.5\ \text{m/s}^2$; with $4\ \text{N}$, at $1\ \text{m/s}^2$; and with $6\ \text{N}$, at $1.5\ \text{m/s}^2$.

1. a) Plot force against acceleration in your exercise book and describe the graph you obtain.
2. b) Deduce the mass of the trolley from the data.
3. c) Predict the acceleration of the trolley if a force of $10\ \text{N}$ were applied to it.

Two lorries with identical engines, producing the same net force of $8000\ \text{N}$, start from rest: one is empty, with a mass of $4000\ \text{kg}$, and the other is loaded, with a mass of $8000\ \text{kg}$.

1. a) Calculate the acceleration of each lorry.
2. b) Calculate the time each lorry takes to reach $20\ \text{m/s}$.
3. c) Using these numbers, explain why a loaded lorry needs more distance to join a motorway.

A full supermarket trolley has a mass of $15\ \text{kg}$ and a net force of $30\ \text{N}$ is applied to it.

1. a) Calculate its acceleration.
2. b) If it starts from rest, what speed does it reach after $3\ \text{s}$?

The engine of a sounding rocket of $3000\ \text{kg}$ produces a net thrust of $12000\ \text{N}$.

1. a) Calculate the acceleration of the rocket.
2. b) If the thrust remains constant, what speed does the rocket gain during the first $10\ \text{s}$?

A car of $1200\ \text{kg}$ tows a caravan of $400\ \text{kg}$ and the engine exerts a net force of $4800\ \text{N}$ on the combination, as shown in the diagram.

1. a) Calculate the acceleration of the car-caravan combination.
2. b) Calculate the force that the towbar exerts on the caravan, keeping in mind that it is the only force accelerating it.

Two perpendicular forces of $6\ \text{N}$ and $8\ \text{N}$ act simultaneously on a wooden block of $5\ \text{kg}$.

1. a) Calculate the magnitude of the net force on the block.
2. b) Calculate the acceleration of the block.
3. c) State how you would determine the direction of this acceleration and justify why it coincides with that of the net force.

Two vans travel at the same speed and brake with the same net braking force, but one is loaded and has twice the mass of the other.

1. a) Compare the braking accelerations of the two vans.
2. b) Compare the times they take to come to a stop.
3. c) Draw a conclusion regarding vehicle load and road safety.

To pull a sledge of $30\ \text{kg}$ across the snow, a force of $90\ \text{N}$ is applied to it and the friction is $30\ \text{N}$. A student calculates the acceleration like this: $a = 90 / 30 = 3\ \text{m/s}^2$.

1. a) Identify the mistake made.
2. b) Calculate the correct acceleration of the sledge.
3. c) Calculate the total force with which it would need to be pulled for the acceleration to actually be $3\ \text{m/s}^2$.