| Step | Derivation/Formula | Reasoning |
|---|---|---|
| 1 | \( d_1 = v_i t + \frac{1}{2} a t^2 \) | Using the kinematic equation for distance travelled \( d \) with initial velocity \( v_i = 0 \), time \( t \), and constant acceleration \( a \). |
| 2 | \( 5 \, \text{m} = \frac{1}{2} a (1 \, \text{s})^2 \) | The distance travelled in the first second is given as \( 5 \, \text{m} \). Substitute \( t = 1 \, \text{s} \) to find the acceleration. |
| 3 | \( 5 = \frac{1}{2} a \) | Simplifying the equation from Step 2. \( 1^2 \) is still \( 1 \), so it simplifies to \( 5 = \frac{1}{2} a \). |
| 4 | \( a = 10 \, \text{m/s}^2 \) | Solving for \( a \) by multiplying both sides of the equation by 2. |
| 5 | \( d_2 = v_i t + \frac{1}{2} a (t + 1)^2 \) | Using the distance formula to calculate the total distance travelled in the first two seconds. Initial velocity \( v_i = 0 \), distance for second time interval \( t_2 = 1 \,\text{s} \). |
| 6 | \( d_2 – d_1 = (v_i + a \cdot 1) \cdot 1 + \frac{1}{2} a \cdot 1^2 \) | Subtract the distance travelled in the first second from the distance travelled in the first two seconds to isolate the distance covered in the second time interval. |
| 7 | \( 15 \, \text{m} – 5 \, \text{m} = 5 + \frac{1}{2} a \cdot 1^2 \) | Given that the distance travelled in the next second is \( 15 \, \text{m} \), the distance covered in the second interval would be \( 15 \, \text{m} – 5 \, \text{m} = 10 \, \text{m} \). |
| 8 | \( 5 + \frac{1}{2} a = 10 \, \text{m} \) | Simplify the above equation \( \frac{1}{2} a \cdot 1 = 10 – 5 = 5 \, \text{m} \). |
| 9 | \( \frac{1}{2} a = 5 \, \text{m} \) | By isolating \( a \) and solving helps in verifying the correctness of previous values calculated. |
| 10 | \( d_3 = \frac{1}{2} a (3)^2 \) | Finally, calculate the total distance travelled in the first three seconds. Using \( t = 3 \, \text{s}\) while keeping other variables the same. |
| 11 | \( d_3 = \frac{1}{2} \cdot 10 \, \text{m/s}^2 \cdot 9 \, \text{s}^2 \) | Substitute the known values for \( a \) and \( t \) into the equation for \( d_3 \). |
| 12 | \( d_3 = 5 \cdot 9 \, \text{m} \) | Simplify by multiplying \( \frac{1}{2} \cdot 10 \) to get \( 5 \) and \( 3^2 = 9 \). |
| 13 | \( d_3 = 45 \, \text{m} \) | The total distance travelled after the 3 seconds is \( 45 \, \text{m} \). |
So, the correct answer is \(\boxed{45 \, \text{m}}\), which corresponds to option (e).
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Which statement is true about the distances the two objects have traveled at time \( t_f \)?
A car is driving to the right at \( 20 \) \( \text{m/s} \). A motorcycle starts \( 30 \) \( \text{m} \) behind the car and is moving at \( 30 \) \( \text{m/s} \) in the same direction.
Toy car W travels across a horizontal surface with an acceleration of \( a_w \) after starting from rest. Toy car Z travels across the same surface toward car W with an acceleration of \( a_z \), after starting from rest. Car W is separated from car Z by a distance \( d \). Which of the following pairs of equations could be used to determine the location on the horizontal surface where the two cars will meet, and why?
A car travels east at a steady \( 30 \) \( \text{m/s} \) for \( 5 \) \( \text{s} \). What is its acceleration during this motion?
You throw a ball straight upward. It leaves your hand at \( 20 \) \( \text{m/s} \) and slows at a steady rate until it stops at the peak. The ball then comes back down, speeding up steadily until it hits the ground with the same speed it left your hand. Draw the velocity vs. time graph or explain it in terms of functions.
A spacecraft accelerates at a rate of \(20.0 \, \text{m/s}^2\).
The first \(10 \, \text{meters}\) of a \(100 \, \text{meter}\) dash are covered in \(2 \, \text{seconds}\) by a sprinter who starts from rest and accelerates with a constant acceleration. The remaining \(90 \, \text{meters}\) are run with the same velocity the sprinter had after \(2 \, \text{seconds}\).
Which pair of graphs represents the same 1-dimensional motion?
You drop a rock off a bridge. When the rock has fallen \( 4 \) \( \text{m} \), you drop a second rock. As the two rocks continue to fall, what happens to their velocities?
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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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