Applications of Integration

For an object in motion along a line with acceleration described by $a(t)=\sin(\pi t)$, how many times between t=0 and t=10 do velocity and acceleration have opposite signs?

If the velocity function of a ball is $v(t) = 5t^2 + 2t$, what is the acceleration of the ball at $t = 2$?

Given that a particle's velocity function is defined as $v(t) = t^4 - 4t^3$, at which value of t does the acceleration function take on its maximum value?

Suppose a particle travels with position $x(t)=t\sin(\pi t)$. At what rate does position change with respect to time at $t=0.52$?

If the velocity function of a car is $v(t) = 3e^t + 2$, what is the acceleration of the car at $t = 1$?

Which graph would represent an object at rest based on its position-time graph $s(t)$?

If the position function of a car is given by $s(t) = 3t^3 - 2t^2 + 5t$, what is the velocity of the car at $t = 2$?

If displacement from origin after a certain duration is represented through $S(T)=\frac{T^3}{T^2+1}$, then during which intervals is the object's instantaneous speed decreasing?

Consider a particle moving along a path. The velocity function of the particle is given by $v(t) = 4t - 3$. What is the position function of the particle?

Considering an initially stationary particle whose acceleration function is given by $a(t)=6t-4$, how should one determine its velocity at $t=3$ using proper calculus methods?