100 m from a traffic light, a motorist sees it to red. If he needs to accelerate at -1 m.s ^2 to stop at the traffic light, what is the initial speed, and how long did he take to stop?
Let initial speed = Vo
Average speed while decelerating = Vo/2
X = 100 m
Time to decelerate to zero =
T = 2 X/Vo
also,
T = Vo/a.
Therefore
2X/Vo = Vo/a
Vo = sqrt(2 a X) = 14.1 m/s
T = 14.1/1 = 14.1 s
To find the initial speed and the time taken to stop, we can use the equations of motion.
Let's consider the equations of motion for linear motion with constant acceleration:
v = u + at (Equation 1)
s = ut + (1/2)at^2 (Equation 2)
where:
- v is the final velocity
- u is the initial velocity
- a is the acceleration
- t is the time taken
- s is the displacement
Given the following information:
- Initial position (s) = 100 m (distance from the traffic light)
- acceleration (a) = -1 m/s^2 (negative sign indicates deceleration)
- Final velocity (v) = 0 m/s (since the motorist has stopped)
Using Equation 2, we can find the initial velocity (u):
s = ut + (1/2)at^2
Substituting the given values:
100 = u * t + (1/2)(-1)t^2
Simplifying:
100 = ut - (1/2)t^2
Now, let's solve for u (initial velocity). Rearrange the equation by isolating u:
u = (1/2)t - (100 / t)
To find the time taken (t), we can solve the equation by using numerical methods such as Newton-Raphson or trial and error. However, without knowing the exact value of t, we cannot calculate the initial velocity (u).
Therefore, we need more information or a specific value for t to provide a precise answer.