Differentiating the position x(t) with respect to time t gives
Velocity v(t) = dx/dt
Acceleration a(t) = dv/dt = d²x/dt²
Example ① Find the velocity and acceleration at t = 2 for the position x(t) = t³ − 3t²。
Solution:v(t) = 3t² − 6t、v(2) = 12 − 12 = 0
a(t) = 6t − 6、a(2) = 6
Solution:v(t) = 3t² − 6t、v(2) = 12 − 12 = 0
a(t) = 6t − 6、a(2) = 6
Key Point
- Velocity = first derivative, acceleration = second derivative
- The sign of velocity indicates the direction of motion (positive means the positive direction, negative means the opposite direction)
Practice Problems
- Find the velocity and acceleration of x(t) = 4sin(2t) at t = π/4。
Solutions and Explanations
- Solution:velocity 0, acceleration −16
v=8cos(2t)、a=−16sin(2t)。