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. a. What is the particle's distance from the origin at t = 0, 2, and 5 s?

Verified step by step guidance

1. Identify the position vector of the particle at different times. The position vector is given by 𝓇 = (5.0î + 4.0ĵ)t². This means that the x-component of the position is 5.0t² and the y-component is 4.0t².
2. Substitute the given times into the position vector equation to find the position at t = 0 s, t = 2 s, and t = 5 s. For example, at t = 0 s, substitute 0 for t in the equation to find the position vector at that time.
3. Calculate the magnitude of the position vector at each time to find the distance from the origin. The magnitude of a vector 𝓇 = xî + yĵ is given by the formula \( \sqrt{x^2 + y^2} \).
4. Apply the magnitude formula to each position vector obtained in step 2. For instance, if the position vector at t = 2 s is (20î + 16ĵ), then the distance from the origin is \( \sqrt{20^2 + 16^2} \).
5. Repeat the calculation for each time point to find the distances at t = 0 s, t = 2 s, and t = 5 s.

Key Concepts

Position Vector

The position vector describes the location of a particle in space relative to a reference point, typically the origin. In this case, the position vector is given as 𝓇 = (5.0î + 4.0ĵ)t² m, indicating that the particle's position changes with time according to the square of time. The components along the x and y axes (î and ĵ) represent the particle's displacement in those directions.

Distance from the Origin

Distance from the origin is the scalar value representing how far a particle is from the reference point (the origin) in space. It can be calculated using the formula for the magnitude of the position vector, which is the square root of the sum of the squares of its components. For the given position vector, this involves evaluating the expression at specific time values to find the distance at t = 0, 2, and 5 seconds.

Time Dependence

Time dependence in physics refers to how a quantity changes over time. In this problem, the position of the particle is explicitly dependent on time, as indicated by the t² term in the position vector equation. Understanding this relationship is crucial for determining the particle's position and distance from the origin at different time intervals.