Topic 2.1 Extension 07

Topic 2.1 Extension 07

Acceleration

1)Recall that the speed or velocity of an object is the time rate of change of the position of the object, or in formula form <v> = x/t and v = dx/dt. What is the difference between <v> and v?

FYI:We define the average acceleration <a of an object as the time rate of change of the speed or velocity of the object, or in formula form

a> = .

2)Suppose the velocity of a particle is given by v = 10t – 5t2, where v is in m/s and t is in seconds. Complete the table below:

t0(s)t1(s)v0(m/s)v1(m/s)v(m/s)t(s)a(m/s2)

2.503.00

2.502.75

2.502.60

2.502.55

2.502.51

FYI:We define the instantaneous acceleration a as the limit as t approaches zero of the average acceleration. Thus

a = = .

FYI:Of course, since we’ll be working in the x, y and z directions, we actually have three instantaneous accelerations defined:

ax = , ay = , az = .

3)Which of one the above values of <a> is closest to the instantaneous acceleration of the particle at t = 2.50 s?

4)Now using the formula v = 10t – 5t2 and the techniques learned in the previous lessons, find the instantaneous acceleration of this particle at any time t (a = dv/dt). a(t) = ______. What is a(2.50)? ______. How does this compare with your answer from 3?

5)Perhaps you recall from kinematics the equation

y(t) = yo + vot + at2,

where y is the position or height of a particle at time t, yo is the original height (a constant), vo is the original speed (a constant) and a is the acceleration (another constant).

a)Find v(t) = dy/dt.

b)Does this formula look familiar?

c)Now find a = dv/dt for the function you obtained in a).

d)Is the acceleration of the particle constant? _____. How do you know?

6)For practice, consider a particle whose y-position is given byy(t) = 250 + 50t - 10t2 where t is in seconds and y is in meters.

a)Find the speed of the particle at any time t.

b)Find the speed of the particle at t = 10 s.

c)Find the acceleration of the particle at any time t.

d)Find the acceleration of the particle at t = 10 s.

e)Is the acceleration of the particle constant?