Uniformly accelerated rectilinear motion

A car starts with an acceleration of $2\ \text{m/s}^2$.

1. a) Explain in your own words what this piece of data means: by how much does its speed increase each second?
2. b) If it starts from rest, what speed will it have after $5\ \text{s}$?

A bicycle starts from rest and accelerates at $3\ \text{m/s}^2$ for $4\ \text{s}$. Calculate its final speed using $v = v_0 + a \cdot t$.

A motorcycle travels at $5\ \text{m/s}$ and accelerates at $2\ \text{m/s}^2$ for $6\ \text{s}$. What speed does it reach?

A car travelling at $20\ \text{m/s}$ brakes with an acceleration of $-4\ \text{m/s}^2$.

1. a) Why is the acceleration negative?
2. b) How long does it take to come to a complete stop?

A bus reduces its speed from $25\ \text{m/s}$ to $15\ \text{m/s}$ in $5\ \text{s}$.

1. a) Calculate its acceleration.
2. b) Explain the meaning of the sign of the result.

A cyclist starts from rest and accelerates at $0,5\ \text{m/s}^2$ for $10\ \text{s}$.

1. a) Calculate the speed reached.
2. b) Calculate the distance covered using $x = x_0 + v_0 \cdot t + \frac{1}{2} a \cdot t^2$.

A car travels at $10\ \text{m/s}$ and accelerates at $2\ \text{m/s}^2$ for $5\ \text{s}$. Calculate the distance it covers in that time.

The position of an object, in International System units, is given by $x = 3 + 2 \cdot t + t^2$. By comparing with $x = x_0 + v_0 \cdot t + \frac{1}{2} a \cdot t^2$:

1. a) Determine its initial position $x_0$, its initial velocity $v_0$ and its acceleration $a$.
2. b) Calculate its velocity when $t = 4\ \text{s}$.
3. c) Calculate its position at that same instant.

A car travelling at $72\ \text{km/h}$ brakes with an acceleration of $-5\ \text{m/s}^2$ until it stops. The $v$-$t$ graph of the braking is shown in the figure.

1. a) Express the initial speed in $\text{m/s}$.
2. b) How long does the braking last?
3. c) What distance does it cover while braking?

A car travels at $90\ \text{km/h}$ when the driver sees an obstacle. It takes $0,8\ \text{s}$ to react (reaction time) and then brakes with $a = -6,25\ \text{m/s}^2$. The two stages of stopping are shown in the figure.

1. a) Express the speed in $\text{m/s}$ and calculate the distance covered during the reaction time.
2. b) Calculate the braking time and the braking distance.
3. c) What is the total stopping distance from the moment the driver sees the obstacle?
4. d) Based on the results, explain why using a mobile phone while driving is so dangerous.

A car travels at a constant speed of $20\ \text{m/s}$ and, as it passes a stationary police car, the police car starts off and pursues it with a constant acceleration of $2\ \text{m/s}^2$.

1. a) Write the position equations for both cars, taking the overtaking point as the origin.
2. b) Calculate the instant at which the police car catches up with the other car. You will obtain two mathematical solutions: explain why one of them is discarded.
3. c) At what distance from the origin does it catch up?
4. d) Calculate the speed of the police car at that instant and compare it with that of the other car.

A cyclist travels at $10\ \text{m/s}$ and, at the moment they pass a stationary moped, the moped starts off and pursues them with an acceleration of $4\ \text{m/s}^2$.

1. a) Write the position equations for both vehicles.
2. b) Solve the resulting quadratic equation to find the time when the moped catches up.
3. c) At what distance from the starting point does this occur?
4. d) What speed is the moped travelling at that moment?

In a training session, runner A passes the starting line at a constant speed of $6\ \text{m/s}$ at the exact instant runner B starts from rest with $a = 1\ \text{m/s}^2$.

1. a) Who is ahead when $t = 8\ \text{s}$? Calculate the position of each runner.
2. b) Calculate the instant when B catches up with A, and the position of the overtake.
3. c) What speed is B travelling at that instant?
4. d) Explain why B, despite starting off slower, ends up catching up with A.