Question

Displacement from a velocity graph Consider the velocity function for an object moving along a line (see figure).
(d) Assuming the velocity remains 10 m/s, for t ≥ 5, find the function that gives the displacement between t = 0 and any time t ≥ 5.
Graph showing velocity in m/s over time in seconds, with a constant velocity of 10 m/s from t=5 seconds onward.

Graph showing velocity in m/s over time in seconds, with a constant velocity of 10 m/s from t=5 seconds onward.

Accepted Answer

Step 1: Understand the problem. The goal is to find the displacement function for t ≥ 5, given that the velocity remains constant at 10 m/s from t = 5 onward. Displacement is the integral of velocity over time. Step 2: Recall the formula for displacement. Displacement is given by the definite integral of the velocity function over the interval of interest. For t ≥ 5, the velocity is constant at 10 m/s, so the integral simplifies. Step 3: Set up the integral. The displacement from t = 0 to any time t ≥ 5 can be expressed as the sum of two parts: (1) the displacement from t = 0 to t = 5, and (2) the displacement from t = 5 to t. For t ≥ 5, the displacement function is: $$ s(t) = \int_{0}^{5} v(t) \, dt + \int_{5}^{t} 10 \, dt . $$Step 4: Evaluate the first integral. From the graph, the velocity function $$ v(t) $$ changes piecewise between t = 0 and t = 5. Break the integral into segments based on the graph: $$ \int_{0}^{5} v(t) \, dt = \int_{0}^{1} 10t \, dt + \int_{1}^{3} 20 \, dt + \int_{3}^{5} (-5t + 35) \, dt . $$Step 5: Evaluate the second integral. For t ≥ 5, the velocity is constant at 10 m/s, so $$ \int_{5}^{t} 10 \, dt = 10(t - 5) . $$Combine the results from Step 4 and Step 5 to express the total displacement function $$ s(t) .$$

Displacement from a velocity graph Consider the velocity function for an object moving along a line (see figure).
(d) Assuming the velocity remains 10 m/s, for t ≥ 5, find the function that gives the displacement between t = 0 and any time t ≥ 5.

Risposta video verificata per un problema simile:

Concetti chiave

Velocity and Displacement Relationship

Integration in Calculus

Piecewise Functions

Displacement from a velocity graph Consider the velocity function for an object moving along a line (see figure).(d) Assuming the velocity remains 10 m/s, for t ≥ 5, find the function that gives the displacement between t = 0 and any time t ≥ 5.