Calculus in Kinematics: Rates of Change

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Questions and Answers

What does the phrase 'rate of change of ______' typically model in calculus?

a derivative

Where is the change in time typically located in a derivative representing a rate of change?

at the bottom

What is the derivative of displacement with respect to time, and what does it represent?

ds/dt, velocity

What are the two derivatives that represent acceleration?

dv/dt and v dv/ds Signup and view all the answers

How do you find the maximum height reached by a projectile?

by finding when velocity is zero in the y-direction Signup and view all the answers

What is the first step in finding the speed of an object given its height function?

differentiate the height function to get a speed function Signup and view all the answers

How do you find the velocity of an object given its height function and a specific time?

by substituting the time into the differentiated height function Signup and view all the answers

What can be obtained by integrating velocity with respect to time?

displacement Signup and view all the answers

Derive the equation `v = u + at` from the given information.

Since the particle moves with a constant acceleration a, we can write dv/dt = a. Integrating both sides with respect to time, we get ∫_u^v▒dv = ∫_0^t▒a dt, which gives v - u = at, hence v = u + at. Signup and view all the answers

Derive the equation `s = ut + 1/2 at^2` from the given information.

From the previous derivation, we have v = u + at. Since ds/dt = v, we can write ds/dt = u + at. Integrating both sides with respect to time, we get ∫_0^s▒ds = ∫_0^t▒(u + at) dt, which gives s = ut + 1/2 at^2. Signup and view all the answers

Derive the equation `v^2 = u^2 + 2as` from the given information.

Since v dv/ds = a, we can write ∫_u^v▒v dv = ∫_0^s▒a ds. Evaluating the integral, we get v^2/2 - u^2/2 = as, which gives v^2 = u^2 + 2as. Signup and view all the answers

A particle moves with a variable acceleration given by dv/dt = 100 - v. Find the time taken for the speed of the particle to increase from 25 m/s to 75 m/s.

Integrating the differential equation, we get ∫_25^75▒dv/(100-v) = ∫_0^t▒dt, which gives -ln(100-v)¦_25^75 = t, hence t ≈ 1.098 s. Signup and view all the answers

Define direct proportionality and give an example.

Direct proportionality occurs when an increase in one quantity results in a proportional increase in the other quantity. This relationship can be expressed as y ∝ x or y = kx, where k is the constant of proportionality. Example: if a car travels at a constant speed, the distance d it travels is directly proportional to the time t it travels, i.e., d = vt. Signup and view all the answers

Solve the differential equation `a = -kv^2` for a car experiencing retardation proportional to the square of its speed.

We can write v dv/ds = -kv^2. Integrating both sides, we get -1/k ∫_10^5▒1/v dv = ∫_0^50▒ds, which gives -1/k ln(v)¦_10^5 = 50, hence k = ln(10/5)/50 = ln(2)/50. Signup and view all the answers

Find the general solution to the differential equation `dC/dt ∝ (C_desired - C)`.

We can write dC/dt = k(C_desired - C). Separating variables and integrating, we get ∫_C_0^C▒dC/(C_desired - C) = ∫_0^t▒k dt, which gives ln(C_desired - C) = kt + c, where c is the constant of integration. Signup and view all the answers

Draw a force diagram for an object falling under the influence of gravity and air resistance, where the air resistance is proportional to the velocity of the object.

The force diagram shows the weight mg acting downwards, and the air resistance F_drag = kv acting upwards. Signup and view all the answers

Derive the differential equation for an object falling under the influence of gravity and air resistance, where the air resistance is proportional to the velocity of the object.

We can write F_net = mg - kv = ma, which gives g - kv/m = dv/dt. Rearranging, we get dv/dt + (k/m)v = g, which is a first-order linear differential equation. Signup and view all the answers

Solve the differential equation `dv/dt + (k/m)v = g` to find v(t) for an object falling under the influence of gravity and air resistance.

This is a first-order linear differential equation, which can be solved using an integrating factor. The general solution is v(t) = (gm/k)(1 - e^(-kt/m)), where m and k are constants. Signup and view all the answers

A 1500 kg car starts from rest and accelerates uniformly. If the engine exerts a constant power of 6000 W, find the velocity of the car after 3 minutes.

v(t) = √(40t/3) and v(180) = 34.65 m/s Signup and view all the answers

The population of a city grows according to the differential equation dP/dt = kP, where P(t) is the population at time t and k is a positive constant. Solve the differential equation to find P(t).

P(t) = P0 e^(kt) Signup and view all the answers

A certain chemical reaction is modeled by the differential equation dC/dt = -kC, where C(t) is the concentration of the chemical at time t and k is a positive constant. If the initial concentration is C0, find the time it takes for the concentration to reduce to one-quarter of its initial value.

t = (1/k) ln(4) Signup and view all the answers

The temperature of a room T(t) changes over time according to the differential equation dT/dt = -k(T - T_env), where T_env is the constant temperature of the environment and k is a positive constant. Find the general solution to the differential equation.

T(t) = T_env + (T0 - T_env) e^(-kt) Signup and view all the answers

A certain radioactive substance decays according to the differential equation dA/dt = -kA, where A(t) is the amount of the substance at time t and k is a positive constant. If the initial amount is A0, find the time it takes for the substance to decay to half of its initial value.

t = (1/k) ln(2) Signup and view all the answers

The rate at which a chemical concentration C(t) changes in a solution is proportional to the difference between the current concentration and a constant desired concentration C_desired. Find the general solution to the differential equation.

C(t) = C_desired + (C0 - C_desired) e^(-kt) Signup and view all the answers

A 2000 kg car starts from rest and accelerates uniformly. If the engine exerts a constant power of 8000 W, find the velocity of the car after 2 minutes.

v(t) = √(40t/3) and v(120) = 40.82 m/s Signup and view all the answers

The concentration of a drug in the bloodstream decreases over time according to the differential equation dC/dt = -kC, where C(t) is the concentration of the drug at time t and k is a positive constant. If the initial concentration is C0, find the time it takes for the concentration to reduce to three-quarters of its initial value.

t = (1/k) ln(4/3) Signup and view all the answers

A particle has acceleration `a = 2t i^→ + t^2 j^→`. Find its velocity after `t` seconds.

<code>v=t^2 i^→+t^3/3 j^→</code> Signup and view all the answers

Find the maximum height of a toy rocket launched at a carnival, given the height function `h(t) = 12t^2 + 4t^2`.

Substitute <code>t = 3</code> into the height function and simplify. Signup and view all the answers

A string has natural length `3m` and elastic constant `5 N/m`. What is the length of the string when it exerts a force of magnitude `30N`?

<code>l = 9 meters</code> Signup and view all the answers

Find the rate of change of the force of attraction, `F = 600/x^2`, with respect to `x`.

<code>dF/dx = -1200/x^3</code> Signup and view all the answers

A particle has displacement `s = t^3/3 i^→ + t^4/12 j^→`. Find its displacement after `12` seconds.

<code>s = 576i^→ + 1728j^→</code> meters Signup and view all the answers

Find the work done in extending an open-coiled spring by `0.1m`, given the natural length `0.3m` and elastic constant `6N/m`.

<code>W = 0.03J</code> Signup and view all the answers

Find the force of attraction between two objects, given the formula `F = 600/x^2`.

Substitute the given value of <code>x</code> into the force formula. Signup and view all the answers

Derive the equation of motion for an object, given its acceleration function `a(t)`.

Integrate the acceleration function with respect to time to find the velocity function, and then integrate the velocity function to find the displacement function. Signup and view all the answers

An elastic string of length `1.5m` and elastic constant `120 N/m` is attached to the ceiling. How far will the string stretch when a `5kg` mass is attached to the end of it?

The string will stretch by <code>0.408 meters</code>. Signup and view all the answers

Find the work done by a variable force, given the force function `F(x)` and the distance `x`.

<code>W = ∫F(x) dx</code> Signup and view all the answers

Study Notes

Rates of Change

When a question mentions "rate of change of ______", we can model it as a derivative.
The rate of change of time will always be at the bottom of the derivative, unless stated otherwise in the question.