Exercises: Physical interpretation of the derivative

The position of a free-falling object is given by the function $s(t) = 100 - 4,9t^2$, where $s$ is measured in metres and $t$ in seconds. Determine the function describing its velocity $v(t)$ and its acceleration $a(t)$.

A train travels along a straight track with an equation of motion $e(t) = 2t^3 - 5t^2 + 8t$, with $e$ in metres and $t$ in seconds. Calculate the velocity of the train at time $t = 2$ s.

The position of a particle as a function of time is $x(t) = t^4 - 3t^2 + 5$. Determine the acceleration of the particle at time $t = 1$.

A hot-air balloon rises such that its height in metres is given by $h(t) = 15t - t^2$. At what time $t$ does its velocity become zero?

The volume of water in a tank changes according to the function $V(t) = 50 + 4t - t^3$, where $V$ is in litres and $t$ in minutes. Find the instantaneous rate of change of volume at time $t = 1$ and state whether the tank is filling or emptying at that moment.

A test car has a velocity given by the function $v(t) = 3t^2 - 12t + 4$ in m/s. Calculate its acceleration at time $t = 3$ s.

The electric charge passing through a cross-section of a wire as a function of time is $q(t) = 2t^2 + 3t$ coulombs. Given that the electric current is the derivative of charge with respect to time, calculate the current at $t = 4$ s.

The height of a projectile in metres is $y(t) = 40t - 5t^2$. Calculate the average velocity in the interval $[1, 3]$ and compare it with the instantaneous velocity at $t = 2$.

An electric scooter moves in a straight line so that its position in meters relative to the starting point is given by $s(t) = t^2 + 2t$, where time $t$ is measured in seconds. To calculate the scooter's velocity exactly $3\text{ segundos}$ after it starts moving, Marcos does the following calculation in his notebook:

«Since $s(3) = 3^2 + 2\cdot 3 = 15\text{ m}$, I divide the position by the time: $v(3) = \frac{15}{3} = 5\text{ m/s}$».

1. a) Explain with reasoning why Marcos's procedure is incorrect and which physical concept he has actually calculated with that operation.
2. b) Use the derivative to find the real instantaneous velocity at $t = 3\text{ s}$, calculate the scooter's acceleration at that same instant, and state whether the vehicle is moving forward or backward.

Before the start of summer, a municipal swimming pool is being drained in a controlled manner. The volume of water remaining in the pool (in cubic metres) after $t$ hours of drainage is described by the function $V(t) = 200 - 20t + 0,5t^2$, valid for the interval $0 \le t \le 10\text{ hours}$.

1. a) Before differentiating, estimate with reasons whether the outflow rate of water will be greater at the beginning of drainage ($t = 1\text{ h}$) or when the pool is almost empty ($t = 9\text{ h}$), based on the pressure exerted by the water column.
2. b) Calculate the average rate of change of the volume between $t = 0$ and $t = 2\text{ hours}$, and use it as a prior estimate of the instantaneous drainage rate.
3. c) Find the outflow rate function using the instantaneous derivative of the volume, calculate its exact value at $t = 1\text{ h}$ and at $t = 9\text{ h}$, and check whether the results confirm your initial estimate.

A toy rocket is launched vertically upwards. Its height in metres after $t$ seconds is $h(t) = 30t - 4,9t^2$. Determine the time it takes to reach its maximum height and calculate the exact value of this height.

The temperature of an industrial oven during a cooling process is given by $T(t) = 200 - 10t + 0,5t^2$ in degrees Celsius, with $t$ in minutes. Calculate the cooling rate (instantaneous rate of change of temperature) at $t = 5$ min and $t = 15$ min. Interpret the physical meaning of the signs obtained.

A vehicle brakes sharply and the distance travelled in metres from the moment the brakes are applied is $d(t) = 24t - 3t^2$. How long does it take for the vehicle to come to a complete stop? What total distance did it travel while braking?

An oil slick on the sea is circular in shape and its radius in metres grows according to the function $r(t) = \frac{20t}{t+2}$, where $t$ is the time in hours. Find the rate at which the radius is growing at time $t = 3$ hours. Does it grow faster at the beginning or after a few hours? Justify your answer using the concept of acceleration or by analysing the derivative.

The position of a harmonic oscillator attached to a spring is $x(t) = 5 \sin(2t)$ in centimetres. Calculate its velocity and acceleration as functions of time. Show that the acceleration is proportional to the position, that is, $a(t) = k \cdot x(t)$, and find the value of the constant $k$.

The production cost of $x$ units of an electronic component is $C(x) = 1500 + 10x - 0,05x^2$ euros. Marginal cost is defined as the derivative of the cost with respect to the number of units. Calculate the marginal cost for $x = 50$ units and explain what this value represents in the context of the factory.

A conical tank empties such that the volume of water inside it is $V(t) = 1000 \left(1 - \frac{t}{20}\right)^3$ litres, for $0 \leq t \leq 20$ minutes. Calculate the outflow rate of water (instantaneous rate of change of volume) at time $t = 10$ minutes.

The position of a particle is given by $s(t) = \frac{t^3}{3} - 3t^2 + 8t$. Determine the time intervals in which the particle moves to the right ($v(t) > 0$) and in which it moves back to the left ($v(t) < 0$).

In a smooth braking test for an autonomous vehicle prototype, the technical team needs to define a rectilinear trajectory using a third-degree polynomial of the form $s(t) = at^3 + bt^2 + ct$, where $s(t)$ is the distance travelled in metres and $t$ the time in seconds elapsed since the start of the test.

The following design conditions are imposed:

1. At the initial instant ($t = 0\text{ s}$), the vehicle travels at a speed of $24\text{ m/s}$.

2. The vehicle comes to a complete stop ($v = 0\text{ m/s}$) $4\text{ seconds}$ into braking ($t = 4\text{ s}$).

3. At that final instant of stopping ($t = 4\text{ s}$), the acceleration must vanish ($a(4) = 0\text{ m/s}^2$) so that the stop is completely imperceptible to the occupants.

1. a) Set up the corresponding system of equations from the derivatives of $s(t)$ and deduce the exact values of the coefficients $a$, $b$, and $c$.
2. b) Justify whether the vehicle moves backwards at any moment during the interval $t \in [0, 4]$ by analysing the sign of the resulting velocity.

A Civia series train of Cercanías de Madrid starts from Atocha station. During the first minute of the journey ($0 \le t \le 60\text{ seconds}$), track sensors record that its position in metres is given by the function $s(t) = 0,5t^2 - \frac{1}{240}t^3$.

1. a) Obtain the analytical expressions for the instantaneous velocity $v(t)$ and acceleration $a(t)$ of the train as a function of time.
2. b) Calculate at what instant the train reaches its top speed in this initial stretch and express this value in kilometres per hour ($\text{km/h}$).
3. c) At the instant $t = 50\text{ s}$, calculate the values of $v(50)$ and $a(50)$. From their signs, interpret physically whether the train is moving forward or backwards, and whether at that instant its speed is increasing or decreasing.

Two race cars, A and B, travel along straight parallel tracks. Their positions relative to the starting line are given by $s_A(t) = t^3 - 6t^2 + 15t$ and $s_B(t) = 2t^2 + 3t$. Determine at what times $t$ both cars are travelling at the same velocity. For these times, calculate and compare the acceleration of each car.

A projectile is launched vertically from a platform. Its height in metres is given by $h(t) = at^2 + bt + c$. We know that the platform is $15$ m above the ground, that the initial launch velocity is $20$ m/s upwards, and that the acceleration due to gravity is constant and equal to $-9,8$ m/s$^2$. Using the relationship between position, velocity, and acceleration via derivatives, find the values of $a$, $b$, and $c$. Next, calculate the maximum height reached.

The velocity of a particle moving in a straight line is $v(t) = \frac{t^2 - 1}{t^2 + 1}$ m/s. Calculate the acceleration function $a(t)$. Determine at what time $t \geq 0$ the acceleration is maximum and calculate this maximum value.

The charge in a capacitor of an electric circuit varies with time according to the expression $q(t) = q_0 \left(1 - e^{-t/RC}\right)$, where $q_0$, $R$, and $C$ are positive constants. The current is $i(t) = q'(t)$. Show that the current decreases exponentially with time and calculate the limiting value of the charge and the current as $t \to +\infty$.

An exploration robot moves along a straight rail. Its position in metres during the first $5$ seconds is given by $x(t) = t^3 - 6t^2 + 9t + 2$. Calculate the actual total distance travelled by the robot in the interval $t \in [0, 5]$. Note that the total distance is not simply $x(5) - x(0)$, but you must analyse the changes in direction of motion using velocity.

Newton's law of cooling states that the rate of change of the temperature $T(t)$ of an object is proportional to the difference between its temperature and that of the surrounding environment $T_a$, that is, $T'(t) = -k(T(t) - T_a)$. Verify that the function $T(t) = T_a + (T_0 - T_a)e^{-kt}$ satisfies this differential equation. If $T_a = 20^\circ$C, $T_0 = 100^\circ$C and the cooling rate at the initial time $t=0$ is $-4^\circ$C/min, calculate the value of $k$.

The power consumed by an electronic device depends on the applied voltage according to the function $P(v) = \frac{v^3}{v^2 + 4}$ milliwatts. The rate of change of power with respect to voltage is called dynamic conductance $G(v) = P'(v)$. Determine for which voltage values $v$ the dynamic conductance is exactly $1$ milliwatt/volt.

A weather balloon rises vertically. Its height in metres is modelled by a piecewise function:

$h(t) = \begin{cases} 2t^2 & \text{if } 0 \leq t \leq 10 \\ at + b - \frac{100}{t} & \text{if } t > 10 \end{cases}$

Determine the values of the parameters $a$ and $b$ so that the position and velocity of the balloon are continuous at time $t = 10$. Once found, calculate the acceleration of the balloon for $t > 10$ and discuss what happens to the velocity in the long term (as $t \to +\infty$).

An olive-growing cooperative in Jaén models the total daily cost (in euros) of bottling $x$ 5-litre jugs of extra virgin olive oil using the function $C(x) = 800 + 15x - 0,01x^2$, for a daily production of up to $500$ jugs. To estimate the economic impact of producing the 201st jug, the finance department considers two alternatives:

• Method A: Calculate the actual incremental cost $\Delta C = C(201) - C(200)$.

• Method B: Approximate that cost using the instantaneous marginal cost by differentiating the function, that is, evaluating $C'(200)$.

1. a) Apply both methods in detail and determine the numerical value obtained with each, as well as the exact difference between them.
2. b) From the definition of the derivative as the limit of the difference quotient with $\Delta x = 1$, reason formally why Method B is an acceptable approximation of Method A.
3. c) If the cooperative were to modernise its facilities and the cost function became non-polynomial, $C(x) = 800 + 12x + \sqrt{x^3 + 100}$, which of the two methods would be more convenient to quickly calculate the impact of successive additional units? Justify your choice by comparing the manual calculation burden against the interpretability of the result.

On the test tracks of the Applus+ IDIADA certification centre in Tarragona, an electric prototype performs a round-trip manoeuvre in a straight line. Its position with respect to the track origin is given by $s(t) = \frac{1}{3}t^3 - 4t^2 + 12t$, where $s$ is measured in metres and $t$ in seconds, over the interval $t \in [0, 8]$.

1. a) Determine at what instants the prototype is momentarily stationary and in what time intervals it moves forward in the positive direction or backwards in the negative direction.
2. b) Obtain the acceleration $a(t)$ and analyse in detail the intervals in which the prototype is speeding up (accelerating) and in which it is slowing down (braking). Pay special attention to the interval $(2, 6)$ and explain how the signs of $v(t)$ and $a(t)$ interact to support your conclusion.
3. c) Calculate the total effective distance travelled by the prototype in the $8\text{ seconds}$ and compare it with the net displacement $s(8) - s(0)$. Explain from the perspective of differential calculus why both quantities do not match.