You are driving to the grocery store at 20 m/s . You are 110m from an intersection when the traffic light turns red. Assume that your reaction time is 0.50s and that your car brakes with constant acceleration.
_ How far are you from the intersection when you begin to apply the brakes?
_ What acceleration will bring you to rest right at the intersection
_ How long does it take you to stop after the light changes.?
v0 = 20m/s
D = dist from int. = 110 - v0 x 0.50
t0 = time to stop
final velocity must equal zero...
0 = v0 - a x t0
t0 = v0/a
D = v0 x t0 - a x t0^2/2
substitute for t0...
D = v0^2/a - a x (v0/a)^2/2
solve for a...
Thank you
To find the distance from the intersection when you begin to apply the brakes, you can use the equation:
distance = initial velocity * time + (1/2) * acceleration * time^2
Here, the initial velocity is 20 m/s, and the time is the reaction time, which is 0.50s. We need to find the distance, so let's substitute the values into the equation:
distance = 20 m/s * 0.50s + (1/2) * acceleration * (0.50s)^2
Using this equation, we can solve for the distance.
To find the acceleration required to bring you to rest right at the intersection, we need to consider that the final velocity would be 0 m/s. We can use the following equation to solve for acceleration:
final velocity^2 = initial velocity^2 + 2 * acceleration * distance
Here, the initial velocity is 20 m/s, the distance is the distance to the intersection, and the final velocity is 0 m/s.
To find the time it takes you to stop after the light changes, we need to consider the time it takes for the car to decelerate to 0 m/s. We can use the following equation to solve for time:
final velocity = initial velocity + acceleration * time
Here, the initial velocity is 20 m/s, and the final velocity is 0 m/s. We can substitute these values into the equation and solve for time.
Let's solve each part one by one.
1. Distance from the intersection when you begin to apply the brakes:
Substitute the values into the equation:
distance = 20 m/s * 0.50s + (1/2) * acceleration * (0.50s)^2
2. Acceleration required to bring you to rest right at the intersection:
Use the equation:
final velocity^2 = initial velocity^2 + 2 * acceleration * distance
3. Time it takes you to stop after the light changes:
Use the equation:
final velocity = initial velocity + acceleration * time
To answer these questions, we can use the equations of motion and the concepts of kinematics. Let's break down each question:
1. How far are you from the intersection when you begin to apply the brakes?
To determine the distance, we need to calculate the distance traveled during the reaction time. The formula to find the distance is:
Distance = Initial velocity x Time + (1/2) x Acceleration x Time^2
Given parameters:
Initial velocity = 20 m/s (speed of the car)
Reaction time = 0.50 s
Using the formula, plug in the values:
Distance = 20 m/s x 0.50 s + (1/2) x Acceleration x (0.50 s)^2
Simplifying:
Distance = 10 m + (0.25) x Acceleration
Therefore, you are (10 m + 0.25 x Acceleration) away from the intersection when you begin to apply the brakes.
2. What acceleration will bring you to rest right at the intersection?
To come to a stop at the intersection, the final velocity should be zero. We can use the formula:
Final velocity^2 = Initial velocity^2 + 2 x Acceleration x Distance
Given parameters:
Final velocity = 0 m/s (car comes to a stop)
Initial velocity = 20 m/s (speed of the car)
Distance = 110 m (distance to the intersection)
Using the formula, plug in the values and solve for acceleration:
0^2 = 20^2 + 2 x Acceleration x 110
0 = 400 + 220 x Acceleration
220 x Acceleration = -400
Acceleration = -400 / 220
Acceleration = -1.82 m/s^2
Therefore, an acceleration of -1.82 m/s^2 will bring you to rest right at the intersection.
3. How long does it take you to stop after the light changes?
To determine the time it takes to stop after the light changes, we can use the equation:
Final velocity = Initial velocity + Acceleration x Time
Given parameters:
Final velocity = 0 m/s (car comes to a stop)
Initial velocity = 20 m/s (speed of the car)
Acceleration = -1.82 m/s^2 (the calculated acceleration)
Using the formula, plug in the values and solve for time:
0 = 20 + (-1.82) x Time
-20 = -1.82 x Time
Time = -20 / -1.82
Time ≈ 10.99 seconds
Therefore, it takes approximately 10.99 seconds for you to stop the car after the light changes.