To find the time taken for the object to travel a distance of 20m with a constant velocity of 5m/s², we need to consider two stages of motion: the initial acceleration phase and the constant velocity phase.
First, let's find the time taken during the initial acceleration phase. We can use the formula:
v = u + at
Where: v = final velocity (5 m/s²) u = initial velocity (10 m/s) a = acceleration (5 m/s²) t = time
Rearranging the equation, we get:
t = (v - u) / a
Substituting the given values:
t = (5 m/s² - 10 m/s) / 5 m/s² t = (-5 m/s) / 5 m/s² t = -1 s
The negative sign indicates that the object would take 1 second to decelerate to a velocity of 5 m/s during the initial acceleration phase.
Now, let's calculate the time taken during the constant velocity phase. We can use the formula:
s = ut + (1/2)at²
Where: s = distance traveled (20 m) u = initial velocity (10 m/s) a = acceleration (5 m/s²) t = time
Rearranging the equation, we get a quadratic equation:
(1/2)at² + ut - s = 0
Substituting the given values:
(1/2)(5 m/s²)t² + (10 m/s)t - 20 m = 0
Simplifying further:
(5/2)t² + 10t - 20 = 0
To solve this quadratic equation, we can use the quadratic formula:
t = (-b ± √(b² - 4ac)) / (2a)
For this equation, a = 5/2, b = 10, and c = -20. Substituting these values into the quadratic formula:
t = (-10 ± √(10² - 4(5/2)(-20))) / (2(5/2)) t = (-10 ± √(100 + 200)) / 5 t = (-10 ± √300) / 5
Calculating further:
t ≈ (-10 ± 17.32) / 5
We have two possible solutions for t:
t₁ ≈ (-10 + 17.32) / 5 ≈ 1.466 s t₂ ≈ (-10 - 17.32) / 5 ≈ -5.066 s
Since time cannot be negative in this context, we discard the negative solution. Therefore, the time taken during the constant velocity phase is approximately 1.466 seconds.
To find the total time taken, we add the time taken during the initial acceleration phase and the time taken during the constant velocity phase:
Total time = 1 s + 1.466 s ≈ 2.466 seconds
Therefore, the object takes approximately 2.466 seconds to travel a distance of 20 meters with an initial velocity of 10 m/s and a constant velocity of 5 m/s².