Newton's Second Law: Force and Acceleration

A remote-controlled car has a mass of $2\ \text{kg}$. What resultant force is required for it to accelerate at $3\ \text{m/s}^2$? Apply $F = m \times a$.

The newton is the SI unit of force: $1\ \text{N}$ is the force that must be applied to a mass of $1\ \text{kg}$ so that it accelerates at $1\ \text{m/s}^2$. From this definition, calculate:

1. a) How many newtons are needed for a mass of $1\ \text{kg}$ to accelerate at $3\ \text{m/s}^2$.
2. b) How many newtons are needed for a mass of $3\ \text{kg}$ to accelerate at $1\ \text{m/s}^2$.

A delivery person pushes a $5\ \text{kg}$ parcel along a smooth floor with a resultant force of $20\ \text{N}$. What acceleration does the parcel acquire? Rearrange $a$ from the expression $F = m \times a$ first and then calculate.

An electric scooter together with its rider accelerates at $3\ \text{m/s}^2$ when the motor exerts a resultant force of $150\ \text{N}$. What is the total mass? Rearrange $m$ from $F = m \times a$ first.

A rucksack has a mass of $4\ \text{kg}$. Calculate its weight on Earth using $P = m \times g$ with $g \approx 10\ \text{m/s}^2$ to get a whole number.

A watermelon has a mass of $2\ \text{kg}$. Calculate its weight on Earth taking the exact value $g = 9.8\ \text{m/s}^2$.

Calculate the weight of a person of $50\ \text{kg}$:

1. a) Taking $g \approx 10\ \text{m/s}^2$.
2. b) Taking the more exact value $g = 9.8\ \text{m/s}^2$.
3. c) What is the difference, in newtons, between the two results?

A gym weight weighs $49\ \text{N}$ (take $g = 9.8\ \text{m/s}^2$).

1. a) What is its mass?
2. b) If you place it on a frictionless floor and push it horizontally with $10\ \text{N}$, what acceleration do you obtain?

You drag a $6\ \text{kg}$ box along the floor by pushing with $40\ \text{N}$, while friction pulls it backwards with $10\ \text{N}$, as shown in the figure.

1. a) What is the resultant force on the box?
2. b) Calculate its acceleration.

A crane lifts a $200\ \text{kg}$ container, as shown in the figure. Take $g \approx 10\ \text{m/s}^2$.

1. a) Calculate the weight of the container.
2. b) If it rises at a constant velocity, what force does the cable pull with? Justify your answer using the first law.
3. c) If the crane wanted the container to accelerate upwards, would it have to pull with more, less, or the same force? Explain using the second law.

A $30\ \text{kg}$ wardrobe is resting on the floor. You push horizontally with $100\ \text{N}$ and it does not move. The figure shows the forces acting on the wardrobe in this situation and in part b.

1. a) If it does not move, what is the resultant force acting on it? What is the friction force then?
2. b) Now you push with $150\ \text{N}$ and the wardrobe starts to slide; while moving, the friction is $120\ \text{N}$. Calculate its initial acceleration.
3. c) Explain why at first it did not move even though you were pushing.

A toy rocket of $0.5\ \text{kg}$ takes off vertically and its engine pushes upwards with $10\ \text{N}$, as shown in the figure. Take $g \approx 10\ \text{m/s}^2$.

1. a) Calculate the weight of the rocket.
2. b) Calculate the upward resultant force acting on it.
3. c) Calculate its initial acceleration at take-off.

You push an empty shopping trolley and a fully loaded one with the same force, as shown in the figure.

Which of the two accelerates more? Explain using Newton's second law, without performing any calculations.

To the same object we first apply a force of $10\ \text{N}$ and then a force of $20\ \text{N}$. If it accelerates at $2\ \text{m/s}^2$ with the first force, with what acceleration will it move with the second? Explain the relationship between force and acceleration when the mass remains constant.

A tennis ball has a mass of $200\ \text{g}$.

1. a) Express its mass in $\text{kg}$.
2. b) Calculate the acceleration produced by a hit with a resultant force of $10\ \text{N}$.

The same force of $12\ \text{N}$ pushes, separately, a $2\ \text{kg}$ chair and a $6\ \text{kg}$ table across a smooth floor.

1. a) Calculate the acceleration of each piece of furniture.
2. b) How many times greater is the acceleration of the chair compared to the table? Relate this to the ratio of their masses.

A $1000\ \text{kg}$ car accelerates from $0$ to $20\ \text{m/s}$ in $5\ \text{s}$.

1. a) Calculate its acceleration (change in velocity divided by time).
2. b) Calculate the resultant force pushing it.

A cyclist and their bicycle have a total mass of $80\ \text{kg}$ and brake from $10\ \text{m/s}$ to $0$ in $4\ \text{s}$.

1. a) Calculate the deceleration (it will be negative, because they are braking).
2. b) What resultant force did the brakes have to exert?

A dog pulls a $25\ \text{kg}$ sledge over snow with a resultant force of $50\ \text{N}$.

1. a) What acceleration does it achieve?
2. b) If two children then get on and the total mass becomes $100\ \text{kg}$, what acceleration does it achieve with the same force? Compare the two results.

A $1200\ \text{kg}$ car accelerates from $0$ to $30\ \text{m/s}$ in $6\ \text{s}$. Its engine can exert a force of $7000\ \text{N}$, as shown in the figure.

1. a) Calculate the acceleration of the car.
2. b) Calculate the resultant force required to achieve that acceleration.
3. c) The difference between the engine force and the resultant force is the total friction (air resistance and tyres): what is its value?

A soccer ball has a mass of $400\ \text{g}$.

1. a) Express its mass in $\text{kg}$.
2. b) During a kick, the cleat exerts an average force of $80\ \text{N}$ on the ball. Calculate the acceleration it receives for the duration of contact.
3. c) You will have obtained a huge acceleration: explain why the ball does not shoot off at infinite velocity, considering how long that contact lasts.

In the laboratory you push a trolley with different forces and measure the accelerations: with $2\ \text{N}$ it accelerates at $1\ \text{m/s}^2$; with $4\ \text{N}$, at $2\ \text{m/s}^2$, and with $6\ \text{N}$, at $3\ \text{m/s}^2$. Look at the graph showing the measurements:

1. a) Deduce the mass of the trolley from any of the data.
2. b) Predict the acceleration if you pushed it with $10\ \text{N}$.
3. c) Describe in your own words the relationship between force and acceleration when the mass remains constant.

On a frozen lake, with negligible friction, you push two blocks of ice, one of $10\ \text{kg}$ and another of $30\ \text{kg}$, with the same force of $60\ \text{N}$ for $2\ \text{s}$ each, as shown in the figure. Both start from rest.

1. a) Calculate the acceleration of each block.
2. b) Calculate the velocity reached by each block after those $2\ \text{s}$.
3. c) Explain why the lighter block ends up moving faster even though the force was the same.

A loaded lorry of $8000\ \text{kg}$ and a car of $1000\ \text{kg}$ are travelling at the same speed. Both have the same maximum braking force available: $4000\ \text{N}$.

1. a) Calculate the maximum deceleration of each vehicle.
2. b) If the car takes $5\ \text{s}$ to stop with maximum braking, how long will the lorry take to stop from the same speed? (Calculate their initial speed first.)
3. c) Draw a conclusion about why lorries need a much greater stopping distance.