Calculus in Kinematics: Rates of Change

a derivative

at the bottom

ds/dt, velocity

<p>dv/dt and v dv/ds</p> Signup and view all the answers

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

<p>differentiate the height function to get a speed function</p> Signup and view all the answers

<p>by substituting the time into the differentiated height function</p> Signup and view all the answers

<p>displacement</p> Signup and view all the answers

<p>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.</p> Signup and view all the answers

<p>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.</p> Signup and view all the answers

<p>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.</p> Signup and view all the answers

<p>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.</p> Signup and view all the answers

<p>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.</p> Signup and view all the answers

<p>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.</p> Signup and view all the answers

<p>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.</p> Signup and view all the answers

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

<p>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.</p> Signup and view all the answers

<p>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.</p> Signup and view all the answers

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

<p>P(t) = P0 e^(kt)</p> Signup and view all the answers

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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