12. A car travels around a horizontal circular track of radius 45m.If the car increases its speed at a constant rate of 1.2m/s2 starting from rest, determine the time needed for it to reach an acceleration of 1.4m/s2. What is its speed at this instant?
To solve this problem, we can use the kinematic equation:
v = u + at
where:
v = final velocity
u = initial velocity (0 in this case, as the car starts from rest)
a = acceleration
t = time
We are given that the car's acceleration increases at a constant rate of 1.2 m/s^2. We need to find the time when the acceleration reaches 1.4 m/s^2.
Let's start solving for the time:
1. First, find the time it takes for the acceleration to increase from 0 to 1.4 m/s^2:
a = u + at
1.4 = 0 + 1.2t
Simplifying the equation, we get:
1.4 = 1.2t
2. Solve for the time (t):
t = 1.4 / 1.2
t = 1.17 seconds (rounded to two decimal places)
So, the car takes approximately 1.17 seconds to reach an acceleration of 1.4 m/s^2.
To find the speed at this instant, we can use the equation:
v = u + at
where:
v = final velocity (what we need to find)
u = initial velocity (0 in this case, as the car starts from rest)
a = acceleration (1.4 m/s^2 in this case)
t = time (1.17 seconds)
Plugging in the values, we can find the speed:
v = 0 + (1.4)(1.17)
v ≈ 1.64 m/s
Therefore, the speed of the car at the instant when the acceleration reaches 1.4 m/s^2 is approximately 1.64 m/s.