Unit 5: Motion and Forces
Summary
Position x of a particle on an x axis
— locates the particle with respect to the origin, or zero point, of the axis.
— is either positive or negative, according to which side of the origin the particle is on, or zero if the particle is at the origin.
— is in the positive direction on an axis when in the direction of increasing positive numbers (to the right).
— is in the negative direction on an axis when in the direction of decreasing positive numbers (to the left).
Displacement
— Δx of a particle is the change in its position: . (x_{2} — x_{1}).
— is a vector quantity.
— is positive if the particle has moved in the positive direction of the x axis and negative if the particle has moved in the negative direction.
— algebraic sign indicates the direction of Δx.
Average Velocity [V_{avg}]
— V_{avg}is a vector quantity where the sign indicates the direction of motion. It is not the actual distance but the displacement of the final position from the initial position. 
V_{avg} = 
(x_{2} — x_{1}) 
——————— 
(t_{2} — t_{1}) 

— when a particle has moved from position x_{1} to position x_{2} during a time interval Δt = t_{2}  t_{1}.
— algebraic sign indicates the direction of v 

— when a graph of x versus t is plotted the average velocity for a time interval is the slope of the straight line connecting the points on the curve.
The slopes V_{avg1} represents the average velocity between A and B for the time interval t_{1}.
The slopes V_{avg2} represents the average velocity between A and C for the time interval t_{2}.
The slopes V_{avg3} represents the average velocity between A and D for the time interval t_{3}.


Average Speed [S_{avg}]
— V_{avg}is a vector quantity where the sign indicates the direction of motion. It is not the actual distance but the displacement of the final position from the initial position. 
S_{avg} = 
total distance 
——————— 
Δt 

—depends on the total distance the particle moves in
that time interval Δt. 
Instantaneous Velocity [v]
v_{} = 

Δx 
lim 
——— 
Δt→0 
Δt 

— Δx is the displacement and Δt is the time interval As Δt→0 the slope approaches the tangent at that point.
— Recall the velocity has both magnitude and direction. The magnitude is the instantaneous speed.
— algebraic sign indicates the direction of v

Average Acceleration [a_{avg}]

— change of velocity Δv over a time interval Δt
— algebraic sign indicates the direction of a_{avg} 
Instantaneous Acceleration [a]
a_{} = 

Δv 
lim 
——— 
Δt→0 
Δt 

— Δv is the change in velocity and Δt is the time interval As Δt→0 the slope approaches the tangent at that point.
— Recall the acceleration has both magnitude and direction.
— algebraic sign indicates the direction of a
— second time derivative of position x(t)
— when a graph of v versus t is plotted the acceleration is the slope of the curve at that points.

Relationship between Position, Velocity and Acceleration

— The slopes of the x vs t graph are velocities for the v vs t graph.
— The slopes of the v vs t graph are accelerations for the a vs t graph.
— Look at t he graph and table for the motion of a car on a short journey. There are four regions.

Time Interval Δt (s) 







01 
13 
38 
89 
910 
Motion 
Stationary 
Speeding Up 
Constant Velocity 
Slowing Down 
Stationary 
Position (m) 
0 → 0 
0 → 7 
7 → 47 
47 → 50 
50 → 50 
Velocity (ms^{1}) 
0 → 0 
0 → 8 
8 → 8 
8 → 0 
0 → 0 
Acceleration (ms^{2}) 
0 → 0 
4 → 4 
0 → 0 
8 → 8 
0 → 0 

Distance zero
Velocity zero
Acceleration zero

Distance increasing
Velocity increasing
Acceleration constant

Distance increasing
Velocity Constant
Acceleration zero

Distance increasing
Velocity decreasing
Acceleration constant
but negative

Distance zero
Velocity zero
Acceleration zero


Equations of Motion (Constant Acceleration)
These are only valid when acceleration is Constant!
v = v_{0} + at  Eq 1
x  x_{0} = v_{0}t + ½at^{2}  Eq 2
v^{2} = v_{02} + 2a(x  x_{0}) Eq3
x  x_{0} = ½(v_{0} + v)t  Eq 4
x  x_{0} = vt  ½at^{2} Eq 5
