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 $v(t)$ of an object is given by $v(t) = 3t^2 - 2t + 1$, what is the expression for the acceleration function $a(t)$?

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

Given a particle's position function on a coordinate plane as $s(t) = t^3 - \frac{9}{2}t^2$, how would you find its speed at time t?

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 acceleration function of the particle is given by $a(t) = 6t + 2$. What is the velocity function of the particle?

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

Consider a particle moving along a path. The position function of the particle is given by $s(t) = t^3 - 2t^2 + 4t - 3$. What is the velocity function of the particle?

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?