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 meters 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 meters 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 meters 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 liters 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 meters is $y(t) = 40t - 5t^2$. Calculate the average velocity in the interval $[1, 3]$ and compare it with the instantaneous velocity at $t = 2$.

A toy rocket is launched vertically upward. Its height in meters 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 traveled in meters 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 meters 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 analyzing the derivative.

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

The production cost of $x$ units of an electronic component is $C(x) = 30\,000 + 200x - 1x^2$ pesos. 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 $V(t) = 1000 \left(1 - \frac{t}{20}\right)^3$ liters, 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 ($v(t) < 0$).

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 meters 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 upward, 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 meters during the first $5$ seconds is given by $x(t) = t^3 - 6t^2 + 9t + 2$. Calculate the actual total distance traveled 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 analyze 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 meters is modeled 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 $t \to +\infty$).