Final answer:
To calculate the velocity at the 5th second, substitute t=5 into the velocity function v(t) = 20t - 5t² m/s to get -25 m/s. The acceleration is found by differentiating the velocity function and at t=5 is -30 m/s², indicating an increasing speed in the opposite direction.
Step-by-step explanation:
To calculate velocity and acceleration at the 5th second, we must analyze the given functions and graphical data that describe the particle's motion. The functional form of the velocity given in Example 3.6 is v(t) = 20t – 5t² m/s. To find the velocity at t = 5 s, we place it into the given velocity function:
v(5) = 20(5) – 5(5)² = 100 – 125 = – 25 m/s.
The negative sign indicates that the direction of the velocity at the 5th second is opposite to the direction at the start. Acceleration can be found by taking the derivative of the velocity function with respect to time, which gives us a(t) = dv/dt = 20 – 10t. So, the acceleration at the 5th second is:
a(5) = 20 – 10(5) = 20 – 50 = – 30 m/s².
This negative acceleration shows that not only is the particle moving in the opposite direction, it is also speeding up in that direction, as the acceleration is increasingly negative.