| Step | Derivation/Formula | Reasoning |
|---|---|---|
| 1 | \( \Delta x = v_{i_x} t \) | Use the horizontal motion formula where \( \Delta x = 32 \, \text{m} \) and solve for \( t \). |
| 2 | \( v_{i_x} = v_i \cos(\theta) \) | Calculate the initial horizontal velocity using the initial velocity and the angle. |
| 3 | \( v_{i_x} = 20 \, \text{m/s} \cdot 0.80 = 16 \, \text{m/s} \) | Substitute \( v_i = 20 \, \text{m/s} \) and \( \cos(37^\circ) = 0.80 \) to find \( v_{i_x} \). |
| 4 | \( 32 \, \text{m} = 16 \, \text{m/s} \cdot t \) | Substitute \( \Delta x = 32 \, \text{m} \) and \( v_{i_x} = 16 \, \text{m/s} \) into the horizontal motion formula. |
| 5 | \( t = \frac{32 \, \text{m}}{16 \, \text{m/s}} \) | Rearrange to solve for \( t \). |
| 6 | \( t = 2 \, \text{s} \) | Calculate the value of \( t \). |
So, the time it takes for the ball to reach the plane of the fence is \( \boxed{2 \, \text{s}} \).
| Step | Derivation/Formula | Reasoning |
|---|---|---|
| 1 | \( y = v_{i_y} t + \frac{1}{2} a t^2 \) | Use the vertical motion equation to find the height \( y \) at \( t = 2 \, \text{s} \). |
| 2 | \( v_{i_y} = v_i \sin(\theta) \) | Calculate the initial vertical velocity using the initial velocity and the angle. |
| 3 | \( v_{i_y} = 20 \, \text{m/s} \cdot 0.60 = 12 \, \text{m/s} \) | Substitute \( v_i = 20 \, \text{m/s} \) and \( \sin(37^\circ) = 0.60 \) to find \( v_{i_y} \). |
| 4 | \( y = 12 \, \text{m/s} \cdot 2 \, \text{s} + \frac{1}{2} (-9.8 \, \text{m/s}^2) \cdot (2 \, \text{s})^2 \) | Substitute \( t = 2 \, \text{s} \), \( v_{i_y} = 12 \, \text{m/s} \), and \( a = -9.8 \, \text{m/s}^2 \) into the vertical motion formula. |
| 5 | \( y = 24 \, \text{m} – 19.6 \, \text{m} \) | Calculate the height \( y \). |
| 6 | \( y = 4.4 \, \text{m} \) | Determine the final value of \( y \) at \( t = 2 \, \text{s} \). |
| 7 | \( y_{\text{final}} – h_{\text{fence}} = 4.4 \, \text{m} – 2.5 \, \text{m} \) | Compare the calculated height with the fence height (\(2.5 \, \text{m}\)). |
| 8 | \( \Delta h = 1.9 \, \text{m} \) | Determine how far above the top of the fence the ball passes. |
So, the ball will pass \( \boxed{1.9 \, \text{m}} \) above the top of the fence.
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Which launch angle gives the greatest horizontal range, assuming level ground and no air resistance?
In the absence of air resistance, a projectile is launched from and returns to ground level and has a range of \( 23 \, \text{m} \). Suppose the launch speed is doubled, and the projectile is fired at the same angle above the ground. What is the new range?
Which statements are not valid for a projectile? Take up as positive.
A officer fires a pistol horizontally toward a target \(120 \,\text{m}\) at a velocity of \(200 \, \text{m/s}\). If the officer aimed directly at the bull’s eye
Projectiles 1 and 2 are launched from level ground at the same time and follow the trajectories shown in the figure. Which one of the projectiles, if either, returns to the ground first, and why?
A cylindrical tank of water (height \( H \)) is punctured at a height \( h \) above the bottom. How far from the base of the tank will the water stream land (in terms of \( h \) and \( H \))? What must the value of \( h \) be such that the distance at which the stream lands will be equal to \( H \)?
During projectile motion (neglecting air resistance), what is the vertical acceleration at the highest point, assuming the initial velocity is upwards in the positive direction?
A baseball is thrown at an angle of 25° relative to the ground at a speed of 23.0 m/s. The ball is caught 42.0 m from the thrower.
Two cannonballs, A and B, are fired from the ground with identical initial speeds, but with \( \theta_A \) larger than \( \theta_B \).
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| Kinematics | Forces |
|---|---|
| \(\Delta x = v_i t + \frac{1}{2} at^2\) | \(F = ma\) |
| \(v = v_i + at\) | \(F_g = \frac{G m_1 m_2}{r^2}\) |
| \(v^2 = v_i^2 + 2a \Delta x\) | \(f = \mu N\) |
| \(\Delta x = \frac{v_i + v}{2} t\) | \(F_s =-kx\) |
| \(v^2 = v_f^2 \,-\, 2a \Delta x\) |
| Circular Motion | Energy |
|---|---|
| \(F_c = \frac{mv^2}{r}\) | \(KE = \frac{1}{2} mv^2\) |
| \(a_c = \frac{v^2}{r}\) | \(PE = mgh\) |
| \(T = 2\pi \sqrt{\frac{r}{g}}\) | \(KE_i + PE_i = KE_f + PE_f\) |
| \(W = Fd \cos\theta\) |
| Momentum | Torque and Rotations |
|---|---|
| \(p = mv\) | \(\tau = r \cdot F \cdot \sin(\theta)\) |
| \(J = \Delta p\) | \(I = \sum mr^2\) |
| \(p_i = p_f\) | \(L = I \cdot \omega\) |
| Simple Harmonic Motion | Fluids |
|---|---|
| \(F = -kx\) | \(P = \frac{F}{A}\) |
| \(T = 2\pi \sqrt{\frac{l}{g}}\) | \(P_{\text{total}} = P_{\text{atm}} + \rho gh\) |
| \(T = 2\pi \sqrt{\frac{m}{k}}\) | \(Q = Av\) |
| \(x(t) = A \cos(\omega t + \phi)\) | \(F_b = \rho V g\) |
| \(a = -\omega^2 x\) | \(A_1v_1 = A_2v_2\) |
| Constant | Description |
|---|---|
| [katex]g[/katex] | Acceleration due to gravity, typically [katex]9.8 , \text{m/s}^2[/katex] on Earth’s surface |
| [katex]G[/katex] | Universal Gravitational Constant, [katex]6.674 \times 10^{-11} , \text{N} \cdot \text{m}^2/\text{kg}^2[/katex] |
| [katex]\mu_k[/katex] and [katex]\mu_s[/katex] | Coefficients of kinetic ([katex]\mu_k[/katex]) and static ([katex]\mu_s[/katex]) friction, dimensionless. Static friction ([katex]\mu_s[/katex]) is usually greater than kinetic friction ([katex]\mu_k[/katex]) as it resists the start of motion. |
| [katex]k[/katex] | Spring constant, in [katex]\text{N/m}[/katex] |
| [katex] M_E = 5.972 \times 10^{24} , \text{kg} [/katex] | Mass of the Earth |
| [katex] M_M = 7.348 \times 10^{22} , \text{kg} [/katex] | Mass of the Moon |
| [katex] M_M = 1.989 \times 10^{30} , \text{kg} [/katex] | Mass of the Sun |
| Variable | SI Unit |
|---|---|
| [katex]s[/katex] (Displacement) | [katex]\text{meters (m)}[/katex] |
| [katex]v[/katex] (Velocity) | [katex]\text{meters per second (m/s)}[/katex] |
| [katex]a[/katex] (Acceleration) | [katex]\text{meters per second squared (m/s}^2\text{)}[/katex] |
| [katex]t[/katex] (Time) | [katex]\text{seconds (s)}[/katex] |
| [katex]m[/katex] (Mass) | [katex]\text{kilograms (kg)}[/katex] |
| Variable | Derived SI Unit |
|---|---|
| [katex]F[/katex] (Force) | [katex]\text{newtons (N)}[/katex] |
| [katex]E[/katex], [katex]PE[/katex], [katex]KE[/katex] (Energy, Potential Energy, Kinetic Energy) | [katex]\text{joules (J)}[/katex] |
| [katex]P[/katex] (Power) | [katex]\text{watts (W)}[/katex] |
| [katex]p[/katex] (Momentum) | [katex]\text{kilogram meters per second (kgm/s)}[/katex] |
| [katex]\omega[/katex] (Angular Velocity) | [katex]\text{radians per second (rad/s)}[/katex] |
| [katex]\tau[/katex] (Torque) | [katex]\text{newton meters (Nm)}[/katex] |
| [katex]I[/katex] (Moment of Inertia) | [katex]\text{kilogram meter squared (kgm}^2\text{)}[/katex] |
| [katex]f[/katex] (Frequency) | [katex]\text{hertz (Hz)}[/katex] |
Metric Prefixes
Example of using unit analysis: Convert 5 kilometers to millimeters.
Start with the given measurement: [katex]\text{5 km}[/katex]
Use the conversion factors for kilometers to meters and meters to millimeters: [katex]\text{5 km} \times \frac{10^3 \, \text{m}}{1 \, \text{km}} \times \frac{10^3 \, \text{mm}}{1 \, \text{m}}[/katex]
Perform the multiplication: [katex]\text{5 km} \times \frac{10^3 \, \text{m}}{1 \, \text{km}} \times \frac{10^3 \, \text{mm}}{1 \, \text{m}} = 5 \times 10^3 \times 10^3 \, \text{mm}[/katex]
Simplify to get the final answer: [katex]\boxed{5 \times 10^6 \, \text{mm}}[/katex]
Prefix | Symbol | Power of Ten | Equivalent |
|---|---|---|---|
Pico- | p | [katex]10^{-12}[/katex] | |
Nano- | n | [katex]10^{-9}[/katex] | |
Micro- | µ | [katex]10^{-6}[/katex] | |
Milli- | m | [katex]10^{-3}[/katex] | |
Centi- | c | [katex]10^{-2}[/katex] | |
Deci- | d | [katex]10^{-1}[/katex] | |
(Base unit) | – | [katex]10^{0}[/katex] | |
Deca- or Deka- | da | [katex]10^{1}[/katex] | |
Hecto- | h | [katex]10^{2}[/katex] | |
Kilo- | k | [katex]10^{3}[/katex] | |
Mega- | M | [katex]10^{6}[/katex] | |
Giga- | G | [katex]10^{9}[/katex] | |
Tera- | T | [katex]10^{12}[/katex] |
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