A car accelerates uniformly from rest. Its acceleration is given by a(t) = 4t m/s², where t is the time in seconds. Find the car’s velocity and position as functions of time.
Step 1: Find the velocity function by integrating the acceleration function with respect to time (t).
v(t) = ∫ a(t)
dt = ∫ 4t dt
Integrating, we get:
v(t) = 2t² + C₁
Since the car starts from rest, its initial velocity v(0) = 0. Therefore, C₁ = 0.
v(t) = 2t²
Step 2: Find the position function by integrating the velocity function with respect to time (t).
x(t) = ∫ v(t) dt = ∫ 2t² dt
Integrating, we get:
x(t) = (2/3)t³ + C₂
Since the car starts at the origin, its initial position x(0) = 0. Therefore, C₂ = 0.
x(t) = (2/3)t³
Now we have the velocity and position functions of the car as a function of time:
v(t) = 2t² (m/s)
x(t) = (2/3)t³ (m)
These functions describe the car’s velocity and position at any given time t during its uniform acceleration.
Activity
A car accelerates uniformly from rest, following the acceleration function a(t) = 5t m/s², where t is the time in seconds. Find the car’s velocity when its position is 10 meters from the starting point.
