Assume the brakes in your car create a constant acceleration of 4.2 m/s2 regardless of your speed. How long does it take to stop if you are initially traveling 50 km/h? What distance do you go while stopping? Repeat the calculations for an initial speed of 100 km/h. What do you notice about the stopping times and distances?
t= 11.9s
To find the time it takes to stop and the distance covered while stopping, we can use the equations of motion.
1. Convert the initial speed from km/h to m/s:
Initial speed = 50 km/h * (1000 m/1 km) * (1 h/3600 s) = 13.89 m/s
2. Use the equation of motion to find the time it takes to stop:
v = u + at
0 = 13.89 m/s + (-4.2 m/s^2) * t
Rearrange the equation:
t = -13.89 m/s / (-4.2 m/s^2)
t = 3.31 seconds
Therefore, it takes approximately 3.31 seconds to stop if you are initially traveling at 50 km/h.
3. Use the equation of motion to find the distance covered while stopping:
s = ut + (1/2)at^2
s = 13.89 m/s * 3.31 seconds + (1/2) * (-4.2 m/s^2) * (3.31 seconds)^2
Simplify and calculate:
s ≈ 45.91 meters
Therefore, while stopping from an initial speed of 50 km/h, you will cover approximately 45.91 meters.
Now let's repeat the calculations for an initial speed of 100 km/h:
1. Convert the initial speed from km/h to m/s:
Initial speed = 100 km/h * (1000 m/1 km) * (1 h/3600 s) = 27.78 m/s
2. Use the equation of motion to find the time it takes to stop:
0 = 27.78 m/s + (-4.2 m/s^2) * t
Rearrange the equation:
t = -27.78 m/s / (-4.2 m/s^2)
t ≈ 6.61 seconds
Therefore, it takes approximately 6.61 seconds to stop if you are initially traveling at 100 km/h.
3. Use the equation of motion to find the distance covered while stopping:
s = ut + (1/2)at^2
s = 27.78 m/s * 6.61 seconds + (1/2) * (-4.2 m/s^2) * (6.61 seconds)^2
Simplify and calculate:
s ≈ 96.35 meters
Therefore, while stopping from an initial speed of 100 km/h, you will cover approximately 96.35 meters.
Observations:
1. The stopping time and distance increase as the initial speed increases. This is because the constant acceleration remains the same, but higher initial speeds require more time and distance to come to a stop.
2. The stopping time for the 100 km/h scenario is roughly twice the stopping time for the 50 km/h scenario, while the stopping distance is also roughly doubled. This indicates a linear relationship between initial speed and stopping time/distance when the braking acceleration is constant.