91Ó°ÊÓ

In kinematics, acceleration is a measure of how quickly the velocity of an object changes with time. For the motorcycle in our problem, the acceleration is given by a quadratic function: \[ a_{x}(t) = A t - B t^2 \]Here, constants \(A\) and \(B\) dictate how acceleration behaves over time. - \(A = 1.50 \, \text{m/s}^3\) shows that initially acceleration increases with time. - \(B = 0.120 \, \text{m/s}^4\) indicates that after a certain point, acceleration starts decreasing as time progresses. Initially, at \(t = 0\), the motorcycle is at rest, meaning both its velocity and position are zero. As time increases, the acceleration initially boosts the velocity until it reaches a maximum point. To find this maximum, we set the acceleration formula to zero, solving for \(t\). This is important because acceleration turning zero will indicate that the velocity stops increasing and might start to decrease if acceleration becomes negative.

Velocity is a central term in kinematics, describing the rate of change of position with respect to time. To determine how velocity varies for our motorcycle, we need to integrate the acceleration function. From the given acceleration equation, integrating over time gives:\[ v(t) = \int (At - Bt^2) dt = 0.5 \, At^2 - 0.3333 \, Bt^3 + C_1\]Since the motorcycle starts from rest \((v(0) = 0)\), \(C_1\) is 0, simplifying our velocity function to:\[ v(t) = 0.5 \, At^2 - 0.3333 \, Bt^3 \]This equation tells us how the velocity changes at any time \(t\). It first increases rapidly due to the positive acceleration term \(0.5 \, At^2\) before it eventually starts to decrease when the negative \(- 0.3333 \, Bt^3\) term dominates.

Integration is a powerful calculus tool used to calculate quantities that aggregate over time, such as velocity from acceleration, and position from velocity. When we integrate the acceleration function of the motorcycle, we obtain the velocity function:\[ v(t) = 0.5 \, At^2 - 0.3333 \, Bt^3\]We then integrate the velocity function to derive the position function:\[ x(t) = \int v(t) dt = 0.167 \, At^3 - 0.0833 \, Bt^4 + C_2\]Again, since the initial position is at the origin \((x(0) = 0)\), \(C_2\) is 0. These integrals connect acceleration, velocity, and position, showing their interdependencies in kinematic analysis. Essentially, integration allows us to understand the complete motion by looking back at how factors accumulate over time. This is indispensable for answering many common questions in kinematics.

Maximum velocity is a valuable concept, as it indicates the peak speed an object reaches under certain conditions. For our motorcycle, this occurs when its acceleration reduces to zero, i.e., when the effect of time causes the acceleration to change its sign. Setting \(a_{x}(t) = At - Bt^2\) to zero and solving gives the critical time:\[ t = \frac{A}{B}\]Substituting this time back into the velocity equation provides the maximum velocity value:\[v_{max} = 0.5 \, A \left(\frac{A}{B}\right)^2 - 0.3333 \, B \left(\frac{A}{B}\right)^3\]The calculation shows us precisely when and how fast the motorcycle goes at its top speed. Employing this method confirms our understanding that velocity will rise until acceleration ceases to be positive due to the increasing dominant effect of \(t^2\). It exemplifies how calculus techniques allow us to not only predict various motion states but also effectively control and optimize them.