objectives. Homework: read 3.1 & 3.2 Exercise: 3.3, 3.5, 3.7. Position, velocity, acceleration vectors. Position vector. (x 2 – x 1 ) i (y 2 – y 1 ) j (z 2 – z 1 ) k. v av =. +. +. ∆t. ∆t. ∆t. Average velocity vector.
Homework: read 3.1 & 3.2
Exercise: 3.3, 3.5, 3.7
Position, velocity, acceleration vectors
If r = bt2i+ ct3j
Where b and c are positive constants, when does the velocity vector make an angle of 45.0o with the x- and y-axes?
v = dr/dt = 2bt i + 3ct2 j
a = i + j + k
a = i + j + k
|a| = √ax2 + ay2 + az2
A projectile is any body that is given an initial velocity and then follows a path determined entirely by the effects of gravitational acceleration and air resistance.
The path of a projectile is called a trajectory
The magnitude of the velocity:
The direction of the velocity:
In y direction:
vy = v0sinθ - gt
y = (v0sinθ)t – ½ gt2
vy2 = (v0sinθ)2– 2gy
In x direction:
vx = v0cosθt
x = (v0cosθ)t
Let’s consider again the skier in example 3.4. What is her acceleration at points G, H, and I after she flies off the ramp? Neglect air resistance.
The acceleration at points G, H, I are the same:
ax = 0; ay = -9.8 m/s2
Example: 3.7 – height and range of a projectile I – a batted baseball
There is no component of acceleration parallel (tangent) to the path; the acceleration vector is perpendicular (normal) to the path and hence directed inward toward the center of the circular path.
arad is always perpendicular to the instantaneous velocity and directed toward the center of the circle. But since v is changing, arad is not constant. arad is greatest at the point in the circle where the speed is greatest.
The component of acceleration that is parallel to the instantaneous velocity is the atan because it is tangent to the circle.
atan is equal to the rate of change of speed.
In uniform circular motion, there is no change in speed, atan = 0
The two quantities are not the same.
Suppose that the particle experiences 4 times the acceleration at the bottom of the loop as it does at the top of loop. Compared to its speed at the top the loop, is its speed the bottom of the loop:
The velocity seen by a particular observer is called the velocity relative to that observe, or simply relative velocity. What is the planes’ speed?
Relative to each other, the planes are almost at rest
Relative to the observers on the ground, the planes are flying at a great speeds.
V(plane to Earth) = V (plane to air) + V (air to Earth)
The acceleration of a(plane to earth) is identical to a(plane to air) because the v(air to earth) is assumed to be constant.