Two protons, each having a speed of 0.935c in the laboratory, are moving toward each other. Determine the total momentum of the two protons in the laboratory.

High-energy particles are observed in laboratories by photographing the tracks they leave in certain detectors; the length of the track depends on the speed of the particle and its lifetime. A particle moving at 0.995c leaves a track 1.25 mm long. What is the proper lifetime of the particle?

A shot fired from a gun with a muzzle velocity of 1200 feet per second is to hit a target 3000 feet away. Determine the minimum angle of elevation of the gun.

A cannon has a muzzle velocity of 800 feet per second. If a cannon ball is to strike a target 10,000 feet downrange, at what angle should the barrel of the cannon be placed? Assume no air resistance or assistance, and assume that the cannon and the target are in the same horizontal plane.

Use the arc length formula from the above Exercise to find the arc length of the curve.

$$x=\cos t+t \sin t, \quad y=\sin t-t \cos t \quad(0 \leq t \leq \pi)$$

a. What is the arc length function associated with $\mathbf{r}(t)=\left<f(t),g(t),h(t))\right>$, where $a\leq t\leq b?$

b. If a curve is parametrized in terms of its arc length, what is the unit tangent vector $\mathbf{T}(s)?$ What is $\mathbf{T}(t)$, where $t$ is not the arc length parameter?

Consider the curve represented by the vector-valued function r(t) = ⟨4(sin t - t cos t), 4(cos t + t sin t), 3/2t²⟩. (a) Write the length of the arc s on the curve as a function of t by evaluating the integral s = ∫_0^t √[x′(u)]² + [y′(u)]² + [z′(u)]² du. (b) Solve for t in the relationship derived in part (a), and substitute the result into the original set of parametric equations. This yields a parametrization of the curve in terms of the arc length parameter s. (c) Find the coordinates of the point on the curve for arc lengths s = √5 and s = 4. (d) Verify that ǁr′(s)ǁ = 1.

Use the arc length formula below, to find the arc length of the curve.

Arc length formula: L = intergal of square root of [(dx/dt) squared + (dy/dt) squared] dt from a to b

x = cos t + t sin t, y = sin t - t cos t, (t in [0, 2pi])

In an RC series circuit, emf $\xi = 12.0$ V, resistance R = 1.40 M$\Omega$, and capacitance C = 1.80 $\mu$F. Find the maximum charge that will appear on the capacitor during charging.

An RC series circuit consists of an AC source, a $33.0 - \mu \text F$ capacitor, and a 30.0-$\Omega$ resistor. If the source emf is given by $\varepsilon = \varepsilon_ { max } \sin ( \omega t )$, where $E _ { \max } = 180.0 \mathrm { V }$ and $\omega = 20.0 \mathrm { s } ^ { - 1 }$, what is the current amplitude?

In the circuit shown in the figure , assume $R=2 \Omega$ and $C=0.1 \mathrm{~F}$. Also, let:

$$v(t)= \begin{cases}0 & \text { for }-\infty<t<0 \\ t & \text { for } 0 \leq t<10 \mathrm{~s} \\ 10 & \text { for } 10 \mathrm{~s} \leq t<\infty\end{cases}$$

Find the energy stored in the capacitor for all time.

Where is the energy stored in a capacitor?

In the circuit shown in the figure , assume $R=2 \Omega$ and $C=0.1 \mathrm{~F}$. Also, let:

$$v(t)= \begin{cases}0 & -\infty<t \leq 0 \\ 2 t & 0 \leq t \leq 1 \mathrm{~s} \\ -(2 t-4) & 1 \leq t \leq 2 \mathrm{~s} \\ 0 & 2 \mathrm{~s} \leq t<\infty\end{cases}$$

Find the energy stored in the capacitor for all time.

Two protons, each having a speed of 0.935c in the laboratory, are moving toward each other. Determine the momentum of each proton in the laboratory.