Figure 3.23 gives the x-t plot of a particle executing one-dimensional simple harmonic motion. (You will learn about this motion in more detail in Chapter14). Give the signs of position, velocity and acceleration variables of the particle at t = 0.3 s, 1.2 s, - 1.2 s.

At t= 0.3 sec,
Negative, Negative, Positive.
At t= 1.2 sec,
Positive, Positive, Negative
At t= -1.2 sec
Negative, Positive, Positive
For simple harmonic motion (SHM) of a particle, acceleration (a) is given by, 
a = – ω²x ;  ω → angular frequency       … (i) 
t = 0.3 s 

In this time interval, x is negative. Thus, the slope of the position-time graph will also be negative. Therefore, both position and velocity are negative.
However, using equation (i), acceleration of the particle will be positive.
t = 1.2 s 

In this time interval, x is positive. Thus, the slope of the x-t graph will also be positive. Therefore, both position and velocity are positive.
However, using equation (i), acceleration of the particle comes to be negative. 
t = – 1.2 s 

In this time interval, x is negative. Thus, the slope of the x-t plot will also be negative. Since both x and t are negative, the velocity comes to be positive.
From equation (i), it can be inferred that the acceleration of the particle will be positive.