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. c. What is the particle's speed at t = 0, 2, and 5 s?

Verified step by step guidance

1. Identify the position vector of the particle, which is given as \( \vec{r} = (5.0 \hat{i} + 4.0 \hat{j}) t^2 \) meters.
2. Calculate the velocity vector \( \vec{v} \) by differentiating the position vector \( \vec{r} \) with respect to time \( t \). The derivative of \( t^2 \) is \( 2t \), so \( \vec{v} = (5.0 \hat{i} + 4.0 \hat{j}) \times 2t = (10.0t \hat{i} + 8.0t \hat{j}) \) meters per second.
3. Find the magnitude of the velocity vector to determine the speed. The magnitude of \( \vec{v} \) is given by \( |\vec{v}| = \sqrt{(10.0t)^2 + (8.0t)^2} \).
4. Simplify the expression for speed: \( |\vec{v}| = \sqrt{100t^2 + 64t^2} = \sqrt{164t^2} = 12.806t \) meters per second.
5. Evaluate the speed at the specific times: at \( t = 0 \) s, \( t = 2 \) s, and \( t = 5 \) s using the expression \( 12.806t \).

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, where î and ĵ are unit vectors in the x and y directions, respectively.

Velocity

Velocity is the rate of change of the position vector with respect to time. It can be found by taking the derivative of the position vector with respect to time. For the given position function, the velocity vector will provide the particle's speed and direction at any instant.

Speed

Speed is a scalar quantity that represents the magnitude of the velocity vector. It is calculated as the square root of the sum of the squares of the components of the velocity vector. To find the speed at specific times, we evaluate the velocity at those times and compute its magnitude.