To calculate the average velocity of a particle moving along the x-axis with the given acceleration, you can use the equations of motion.
The acceleration of the particle is given by:
a = ao(1 - t/T)
Integrating the above equation with respect to time will give you the velocity of the particle:
v = ∫(a) dt v = ∫(ao(1 - t/T)) dt v = ao∫(1 - t/T) dt
Now, integrating the right side of the equation:
v = ao∫(1 - t/T) dt v = ao[t - (t^2 / 2T)] + C v = ao[t - t^2 / (2T)] + C
Here, C is the constant of integration. To determine C, you would need initial conditions or further information about the particle's motion.
To calculate the average velocity, you need to find the displacement of the particle over a given time interval (let's say from t1 to t2) and divide it by the duration of the interval:
Average velocity (Vavg) = (x2 - x1) / (t2 - t1)
However, since we have the equation for velocity in terms of time (v = ao[t - t^2 / (2T)] + C), we can find the displacement by integrating the velocity equation with respect to time:
x = ∫(v) dt x = ∫[ao(t - t^2 / (2T))] dt x = ao[ (t^2 / 2) - (t^3 / (3 * T))] + Ct + D
Here, D is the constant of integration. As before, we would need additional information to determine the constants C and D.
Now, with the displacement equation, we can find the average velocity by plugging in the values of t2 and t1:
Average velocity (Vavg) = (x2 - x1) / (t2 - t1) Vavg = [ao( (t2^2 / 2) - (t2^3 / (3 * T))) + Ct2 + D - ao( (t1^2 / 2) - (t1^3 / (3 * T))) - Ct1 - D] / (t2 - t1)
Simplifying the equation will give you the expression for the average velocity in terms of the given values of t1 and t2.