12PHYS - Mechanics
Finn Le Sueur
2024
Head up your book a bold title “Speed and Acceleration” and write the date and te whāinga ako below.
In pairs with your whiteboard write down your answer and get ready to hold it up!
Bob because he ran \(100m\) in the shortest time.
Aaron because he ran the furthest in \(1 minute\).
\[ \begin{aligned} v &= \frac{\Delta d}{\Delta t} \cr d &= \text{total distance travelled (meters)} \cr t &= \text{time (seconds)} \cr v &= \text{speed} \end{aligned} \]
Write this equation in your book and give the unit for each letter in the equation.
Ash runs \(315m\) in \(45s\). Calculate his average speed in meters per second.
\[ \begin{aligned} & && \text{Knowns} \cr & && \text{Unknowns} \cr & && \text{Formula} \cr & && \text{Sub and Solve} \end{aligned} \]
Ash runs \(315m\) in \(45s\). Calculate his average speed in meters per second.
\[ \begin{aligned} d &= 315m, t = 45s \cr v &= ? \cr v &= \frac{\Delta d}{\Delta t} \cr v &= \frac{315}{45} \cr v &= 7ms^{-1} \end{aligned} \]
Pātai: In pairs, convert the speed of an airplane to meters per second.
\[ \begin{aligned} v &= \frac{1100km}{hr} \cr &= \frac{1100km \times 1000}{60 \times 60} \cr &= \frac{1100000}{3600} = 305.56ms^{-1} \end{aligned} \]
\[ \begin{aligned} & && \text{Knowns} \cr & && \text{Unknowns} \cr & && \text{Formula} \cr & && \text{Sub and Solve} \end{aligned} \]
A car is moving at a speed of \(10ms^{-1}\). How far does the car travel in \(12s\)?
\[ \begin{aligned} v &= 10ms^{-1}, t=12s && \text{Knowns} \cr d &= ? && \text{Unknowns} \cr v &= \frac{d}{t} && \text{Formula} \cr 10 &= \frac{d}{12} && \text{Sub and Solve} \cr 10 \times 12 &= d = 120m \end{aligned} \]
A man is running at a speed of \(4ms^{-1}\). How long does he take to run \(100m\)?
\[ \begin{aligned} v &= 4ms^{-1}, d=100m \cr t &= ? \cr v &= \frac{d}{t} \cr 4 &= \frac{100}{t} \cr 4 \times t &= 100 \cr t &= \frac{100}{4} = 25s \end{aligned} \]
Velocity may refer to average velocity or instantaneous velocity.
The formula \(v = \frac{d}{t}\) can only be used to calculate average velocity or when the velocity is constant.
To find the instantaneous velocity you must use calculus. This is not something we learn in PHY201.
The rate of change in speed
\[ \begin{aligned} a &= \frac{\Delta v}{\Delta t} \cr \Delta v &= \text{ change in speed} \cr t &= \text{ time (s)} \cr a &= \text{ acceleration} \end{aligned} \]
meters per second squared OR meters per second per second
For example, \(a=12ms^{-2}\) means that the velocity is increased by \(12ms^{-1}\) every second.
| t (s) | 0 | 1 | 2 | 3 |
|---|---|---|---|---|
| v (m/s) | 0 | 12 | 24 | 36 |
This is the difference between the initial and the final value.
\[ \begin{aligned} & \Delta = final - initial \cr & \text{e.g. }\Delta v = v_{f} - v_{i} \end{aligned} \]
A man initially walking at \(2.0ms^{-1}\) notices that his house is on fire so he speeds up to \(4ms^{-1}\) in \(1.3s\).
\[ \begin{aligned} & v_{f} = 4ms^{-1}, v_{i} = 2ms^{-1} && \text{Knowns}\cr & \Delta v = ? && \text{Unknowns}\cr & \Delta v = v_{f} - v_{i} && \text{Formula}\cr & \Delta v = 4 - 2 =2ms^{-1} && \text{Sub and Solve} \end{aligned} \]
\[ \begin{aligned} & \Delta v = 2ms^{-1}, t = 1.3s && \text{Knowns} \cr & a = ? && \text{Unknowns} \cr & a = \frac{\Delta v }{t} && \text{Formula} \cr & a = \frac{2}{1.3} = 1.54ms^{-2} && \text{Sub and Solve} \end{aligned} \]
A cyclist who has been travelling at a steady speed of \(4ms^{-1}\) starts to accelerate. If he accelerates at \(2.5ms^{-2}\), how long will he take to reach a speed of \(24ms^{-1}\)?
K,U,F,S,S
\[ \begin{aligned} & v_{i} = 4ms^{-1}, v_{f} = 24ms^{-1}, a = 2.5ms^{-2} \cr & t = ? && \text{Unknowns} \cr & a = \frac{\Delta v}{t} && \text{Formula} \cr & t = \frac{\Delta v}{a} && \text{Rearrange} \cr & t = \frac{v_{f}-v_{i}}{a} && \text{Expand $\Delta$v} \cr & t = \frac{24 - 4}{2.5} && \text{Substitute} \cr & t = 8s && \text{Solve} \end{aligned} \]
Whakatika 1
\[ \begin{aligned} & v_{i} = 12.7ms^{-1}, a = 1.3ms^{-2}, t = 60s && \text{Knowns} \cr & v_{f} = ? && \text{Unknowns} \cr & a = \frac{v_{f} - v_{i}}{t} && \text{Formula} \cr & a \times t = v_{f} - v_{i} && \text{Rearrange for final v} \cr & v_{f} = (a \times t) + v_{i} \cr & v_{f} = (1.3 \times 60) + 12.7 = 90.7ms^{-1} && \text{Sub and solve} \end{aligned} \]
Whakatika 2
\[ \begin{aligned} & a = -1.8ms^{-2}, t = 9.4s, v_{f} = 0ms^{-1} && \text{Knowns} \cr & v_{i} = ? && \text{Unknowns} \cr & a = \frac{v_{f} - v_{i}}{t} && \text{Formula} \cr & a \times t = v_{f} - v_{i} && \text{Rearrange for initial v} \cr & v_{i} = v_{f} - (a \times t) \cr & v_{i} = 0 - (-1.8 \times 9.4) = 16.92ms^{-1} && \text{Sub and solve} \end{aligned} \]
Discuss with the person next to you, the relevance of the positive and negative signs.