W1 Motion and Vectors

Displacement vs Distance

Displacement

• $\boxed{\Delta x = x_{f}-x_{i}}$
  • $\Delta x$: Displacement
  • $x_f$: final position
  • $x_i$: initial position
• change in position of an object
• can be positive or negative

Distance

• the magnitude or size of displacement between two positions
• distance traveled is the total length of the path traveled between two positions
• distance is either more than ro equal to displacement

Scalar VS Vector

	Scalar	Vector
Magnitude	✅	✅
Direction	✅	❌

Time, Velocity, and Speed

Time

• $\Delta t = t_f-t_i$
• $\Delta t$: change in time / elapsed time
• $t_f$: time at the end of motion
• $t_i$: time at the beginning of motion

Velocity

• $\boxed{v_{ave}=\frac{\Delta x}{\Delta t}=\frac{x_f-x_i}{t_f-t_i}}$
  • $v_{ave}$: average velocity
  • $\Delta x$: displacement
  • $\Delta t$: elapsed time
• how fast an object travels in a given direction
• vector

Instantaneous Velocity

• $\boxed{v=\lim_{\Delta t\rightarrow 0}\frac{\Delta x}{\Delta t}=\frac{dx}{dt}}$
• ‘velocity’ usually means ‘instantaneous velocity’ unless specified otherwise

Speed

• how fast/how much distance travelled in a given time interval
• $\textmd{Speed}=\frac{\textmd{distance travelled in a given time interval}}{\textmd{time interval}}$
• scalar

Acceleration

• $\boxed{a_{ave}=\frac{\Delta v}{\Delta t}=\frac{v_f-v_i}{t_f-t_i}}$
  • $a_{ave}$: average acceleration
  • $\Delta v$: change in velocity
  • $\Delta t$: change in time
• vector

Kinematic Equations (Constant Acceleration)

1. $v=\frac{\Delta x}{\Delta t}=\frac{v_f-v_i}{2}$
2. $v_f = v_i + a\Delta t$
3. $\Delta x = x_i + v_i\Delta t + \frac{1}{2}a(\Delta t)^2$
4. $\Delta x=\frac{1}{2}(v_i+v_f)\Delta t$
5. $2a\Delta x=v_f^2-v_i^2$

Graphs

Falling Objects

Gravity

• $\boxed{g=-9.80 m/s^2}$ (upward is position; downward is negative)

Kinematic Equations for Objects in Free-Fall:

1. $v_f=v_i+gt$
2. $y_f=y_i+v_i\Delta t+\frac{1}{2}g(\Delta t)^2$
3. $2a\Delta y=v_f^2-v_i^2$

Ball dropped from a tower

• Question Suppose that a ball is dropped ($v_0=0$) from atower. How far will it have fallen after a time:
  1. $t_1=1.00s$
  2. $t_2=2.00s$
  3. $t_3=3.00s$
• Take $y$ as positive downward
• so acceleration is $a=g=+9.80m/s^2$
• $v_0=0m/s$
• $y_0=0m$
• Use Kinematic Equations for Objects in Free-Fall (2)
  • $y_f=y_i+v_i\Delta t+\frac{1}{2}g(\Delta t)^2$

1. $t_1=1.00s$
   • $y_1=0m+0m/s(1s)+\frac{1}{2}(9.8m/s^2)(1s)^2$
   • $\therefore y_1=4.9m$
2. $t_2=2.00s$
   • $y_1=0m+0m/s(2s)+\frac{1}{2}(9.8m/s^2)(2s)^2$
   • $\therefore y_1=19.6m$
3. $t_3=3.00s$
   • $y_1=0m+0m/s(3s)+\frac{1}{2}(9.8m/s^2)(3s)^2$
   • $\therefore y_1=44.1m$

Ball thrown down from a tower

• Question Suppose that a ball is thrown downward with an initial velocity of $3.00m/s$
  1. What would be it’s position after 1s, 2s, 3s?
  2. What would its speed be after 1s, 2s?

Ball thrown upward

• Question A person throws a ball upward into the air with an initial velocity of $15.0m/s$. calculate how high it goes. Ignore air resistance.
• Take $y$ as positive upward
• so $a=-g=-9.8m/s^2$
• As the ball rises, its speed decreases until it reaches the highest point, where its speed is zero for an instant; then it descends with increasing speed
• At $t=0s$:
  • $y_0=0m$
  • $v_0=15m/s$
  • $a=-9.8m/s^2$
• At $t=max$
  • $v_m=0$
  • $a=-9.8m/s^2$
• Use Kinematic Equation for Objects in Free-Fall (3)
  • $2a\Delta y=v_f^2-v_i^2$
• $2g\Delta y=v_m^2-v_0^2$
• $2(-9.8m/s)(y_m-0m)=(0m/s)^2-(15m/s)^2$
• $y_m=11.48959184m$
• $\approx 11.5m$ <3sf>

Ball thrown upward at edge of cliff

Vectors

Projectile Motion

⚠️ EQUATIONS OF PROJECTILE MOTION

• $v_f=v_i+a\Delta t$
• $\Delta x=v_i\Delta t+\frac{1}{2}a(\Delta t)^2$
• $\Delta x=\frac{1}{2}(v_i+v_f)\Delta t$
• $2a\Delta x=v_f ^2-v_i ^2$

Horizontal motion ($x$ direction) [don’t need to remember]

• $a_x=0$
• $(v_x)_f=(v_x)_i$
• $x_f=x_i+v_i\Delta t$

Vertical motion ($y$ direction) [don’t need to remember]

• $a_y=-g$
• $(v_y)_f=(v_y)_i-g\Delta t$
• $y_f=y_i+(v_y)_i\Delta t-\frac{1}{2}g(\Delta t)^2$

Textbook Problems

• Ch2: 3,6,9,10,15*,16*,18,20,25,31,35*,38,49*,57
• Ch3: 3,4,9,13,18,19,26,27,32,35*