Using the Derivative

AP Physics CMrs. Coyle

http://www.ima.umn.edu/~arnold/graphics.html

Instantaneous Velocity

v = lim xt 0t

v = dxdt

or

Instantaneous Acceleration

or

a = lim vt 0t

a = dvdt

Using the limit to calculate instantaneous acceleration.

• Example 1:The velocity of a particle is given by v= -t2 + 2 (t is in sec).

Find the instantaneous acceleration at t= 4s (using the limit).

Answer: -8 m/s

Evaluating the derivative of a polynomial.For y(x) = axn

dy = a n xn-1

dx

-Apply to each term of the polynomial.-Note that the derivative of constant is 0.

Using the derivative to calculate instantaneous acceleration.

• Example 2:The velocity of a particle is given by v= -t2 + 2 (t is in sec).

Find the instantaneous acceleration at t= 4s (using the derivative).

Answer: -8 m/s

Example 3:

A particle’s position is given by the expression x= 4-t2 + 2t3 (t is in sec).

Find for t= 5s :a) Its positionb)Its velocityc) Its acceleration

Answer: a) 229m, b) 140 m/s, c) 58 m/s2

Example 4An object follows the equation of motion x= 3t2 -10t +5.a) At what time(s) is its position equal to zero?b) At what time is its velocity equal to zero? Hint: Remember for a quadratic equation ax² + bx + c = 0 , the roots are: Answer: a) 0.62sec and 2.7sec, b) 1.7sec