The position of a particle as a function of time is given by 𝓇 = ( 5.0î ＋4.0ĵ )t² m where t is in seconds. b. Find an expression for the particle's velocity v as a function of time.

Verified step by step guidance

1. Identify the given position function of the particle, which is 𝓇 = (5.0î + 4.0ĵ)t² meters.
2. Recognize that velocity is the first derivative of the position function with respect to time. Thus, to find the velocity function, differentiate the position function 𝓇 with respect to time t.
3. Apply the power rule of differentiation to each component of the position vector. For a function of the form Ct², where C is a constant, the derivative is 2Ct.
4. Differentiate the x-component (5.0t²) and the y-component (4.0t²) of the position vector separately. The derivative of 5.0t² with respect to t is 10.0t, and the derivative of 4.0t² with respect to t is 8.0t.
5. Combine the differentiated components to form the velocity vector v as a function of time: v = (10.0tî + 8.0tĵ) meters per second.

Key Concepts

Position Vector

The position vector describes the location of a particle in space as a function of time. In this case, the position vector 𝓇 = (5.0î + 4.0ĵ)t² m indicates that the particle's position changes quadratically with time, with components in the x and y directions represented by the coefficients of î and ĵ, respectively.

Velocity

Velocity is the rate of change of the position vector with respect to time. It is a vector quantity that indicates both the speed and direction of the particle's motion. To find the velocity as a function of time, one must differentiate the position vector with respect to time, yielding a new vector that describes how the position changes over time.

Differentiation

Differentiation is a fundamental concept in calculus that involves finding the derivative of a function. In physics, it is used to determine rates of change, such as velocity from position. By applying differentiation to the position vector with respect to time, we can derive the expression for velocity, which provides insight into the particle's motion at any given moment.