A driver of a car going 95.0 km/h suddenly sees the lights of a barrier 39.0 m ahead. It takes the driver 0.95 s before he applies the brakes, and the average acceleration during braking is -10.0 m/s2.
What is the maximum speed at which the car could be moving and not hit the barrier 39.0 m ahead? Assume the acceleration rate doesn't change.
Ignore the 95.0 km/h speed of the driver. He may hit the barrier. They want you to calculate the speed V for which the stopping distance would be exactly 39.0 m.
The stopping distance will be
V*0.95 + (V/{a})*(V/2) = 39.0 m
Solve for V.
The second term, V^2/(2|a|)
is the distance travelled with brakes applied, It equals the average speed times the time spent braking. The first term, V/0.95s, it the distance travelled before the brakes are applied,
Can you go through and show all work to the finish?
To find the maximum speed at which the car could be moving and not hit the barrier, we can use the following steps:
Step 1: Convert the given values to SI units
The car's initial velocity, v₀ = 95.0 km/h
Convert 95.0 km/h to m/s: (95.0 km/h) * (1 h / 3600 s) * (1000 m / 1 km) = 26.39 m/s
Step 2: Calculate the distance covered during the reaction time
The time taken to apply the brakes, t = 0.95 s
During this time, the car covers a distance, d₁ = v₀ * t
d₁ = (26.39 m/s) * (0.95 s) = 25.07 m
Step 3: Calculate the distance covered during braking using the average acceleration
The average acceleration during braking, a = -10.0 m/s^2
The distance covered during braking, d₂, can be calculated using the formula:
d₂ = (v² - u²) / (2 * a)
where v = final velocity, u = initial velocity
d₂ = (0 - (26.39 m/s)²) / (2 * (-10.0 m/s^2))
d₂ = (-695.52 m²/s²) / (-20.0 m/s²)
d₂ = 34.78 m
Step 4: Total distance covered before hitting the barrier
The total distance covered, d = d₁ + d₂
d = 25.07 m + 34.78 m = 59.85 m
Step 5: Compare the total distance covered with the distance to the barrier
If the total distance covered, d, is greater than or equal to the distance to the barrier, the car will hit the barrier. Otherwise, it will miss the barrier.
If d >= 39.0 m, the car will hit the barrier.
If d < 39.0 m, the car will miss the barrier.
In this case, the total distance covered, 59.85 m, is greater than the distance to the barrier, 39.0 m. Therefore, the car would hit the barrier.
Hence, the maximum speed at which the car could be moving and not hit the barrier is 95.0 km/h or 26.39 m/s.
To find the maximum speed at which the car could be moving and not hit the barrier, we need to calculate the distance the car could travel while applying the brakes to come to a stop before reaching the barrier.
We know the initial speed of the car is 95.0 km/h, which can be converted to m/s by dividing by 3.6:
Initial speed = 95.0 km/h = (95.0 * 1000) / (60 * 60) m/s = 26.4 m/s
We also know the acceleration during braking is -10.0 m/s^2.
The time it takes to apply the brakes before coming to a complete stop is given as 0.95 s.
Using the equation of motion, we can calculate the distance:
Distance = Initial speed * Time + 0.5 * Acceleration * Time^2
Distance = (26.4 m/s * 0.95 s) + 0.5 * (-10.0 m/s^2) * (0.95 s)^2
Distance = 25.08 m - 4.75 m
Distance = 20.33 m
Therefore, the maximum speed at which the car could be moving and not hit the barrier is when the distance traveled under braking is equal to or greater than the distance to the barrier. In this case, it would be a speed at which the car could come to a stop within 39.0 m.
So, the maximum speed at which the car could be moving and not hit the barrier is 26.4 m/s.