The acceleration of a particle is given by a=-k t^2 where a...

The acceleration of a particle is given by $a=-k t^{2}$ where $a$ is in meters per second squared and the time $t$ is in seconds. If the initial velocity of the particle at $t=0$ is $v_{0}=12 \mathrm{m} / \mathrm{s}$ and the particle takes 6 seconds to reverse direction, determine the magnitude and units of the constant $k .$ What is the net displacement of the particle over the same 6 -second interval of motion?

Key Concepts

Determining the Constant with Direction Change

A key element in problems involving motion is recognizing the point of reversal of direction, which occurs when the velocity becomes zero. By setting the velocity function equal to zero at the given time, one can solve for unknown constants such as k. This approach demonstrates how critical points in motion, such as the instant of direction reversal, can be used to extract specific physical parameters via the underlying mathematical relationships.

Displacement Calculation from Velocity

Displacement is determined by integrating the velocity function with respect to time over a specified time interval. The definite integral of velocity yields the net displacement of the particle, regardless of the changes in direction during motion. This concept underscores the relationship between the rate of change of position and the total change in position over a given period.

Application of Initial Conditions

Initial conditions are critical in solving differential equations in physics because they determine the unique solution corresponding to the physical scenario being considered. In this problem, the initial velocity is specified, which is used to determine the constant of integration when obtaining the velocity function from the acceleration function. The initial condition ensures that the kinematic equations correctly model the motion of the particle at the start of observation.

Acceleration and its Integration

In kinematics, acceleration is defined as the rate of change of velocity with respect to time. When acceleration is given as a function of time, integration is used to obtain the velocity function. This process involves integrating the acceleration function and applying the appropriate initial conditions to determine the constant of integration, ensuring that the resulting velocity accurately reflects the particle's motion from its starting state.

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