Angelo is driving at 15.0 m/s when he saw a stalled car 75 m ahead. At what rate must he slow down his car so that it will stop 0.5 m before hitting the stalled car? His reaction time is 1.2 s.
in 1.2 seconds, the car will travel 18m, so the stopping distance is only
75-18-0.5 = 56.5m
So now we need to find a and t such that
15+at = 0
15t + 1/2 at^2 = 56.5
15(15/a) - 1/2 a (15/a)^2 = 56.5
a = -2 m/s^2
Well, Angelo certainly needs to hit the brakes before he meets his new friend, the stalled car. Let's calculate how much of a speed reduction he needs to make.
First, let's take into account his reaction time. If Angelo's reaction time is 1.2 seconds, that means he'll need some extra distance to react and start slowing down. Luckily, the speed of light is not a limiting factor here, so we only need to consider his speed.
In 1.2 seconds, Angelo will travel a distance of 15.0 m/s multiplied by 1.2 seconds, which is 18 meters. This means that Angelo will cover 18 meters before he even starts to slow down. So, now the initial distance between Angelo and the stalled car is 75 m - 18 m = 57 meters.
Now, we need Angelo to stop 0.5 meters before hitting the stalled car. This means he needs to reduce his speed in such a way that he only covers this 0.5 meters distance after reacting.
Given that we now have a new distance of 57 meters and a final distance of 0.5 meters, we divide the change in distance by his reaction time: (57 m - 0.5 m)/1.2 s ≈ 47 m/s.
So, Angelo needs to slow down by approximately 47 meters per second (or approximately 169.2 kilometers per hour) to make sure he stops 0.5 meters before hitting the stalled car. That's one way to put the brakes on, huh?
To determine the rate at which Angelo must slow down his car, we can use the equations of motion.
Let's break down the problem step by step:
Step 1: Find the time it takes for Angelo to react
Given that Angelo's reaction time is 1.2 s, we know that he takes 1.2 s before he starts to slow down the car. This means that the actual distance he has to cover during the reaction time is (15.0 m/s * 1.2 s) = 18.0 m.
Step 2: Calculate the distance required to stop
The distance required to stop the car is the sum of the distance covered during the reaction time and the distance from the stalled car. Given that he needs to stop 0.5 m before hitting the stalled car, the total stopping distance becomes (75 m + 0.5 m) = 75.5 m.
Step 3: Determine the actual stopping distance
The actual stopping distance is the distance covered during the reaction time plus the distance covered while decelerating. Let's denote this distance as D.
D = 18.0 m + deceleration_distance
Step 4: Apply the equation of motion for the actual stopping distance
The equation for the actual stopping distance, assuming constant acceleration, is:
D = (initial velocity * time) + (0.5 * acceleration * time^2)
Step 5: Substitute the values into the equation
Since we're looking for the deceleration rate, we can rearrange the equation to solve for acceleration:
acceleration = (D - (initial velocity * time)) / (0.5 * time^2)
acceleration = (75.5 m - (15.0 m/s * 1.2 s)) / (0.5 * (1.2 s)^2)
Step 6: Calculate the acceleration
By substituting the values into the equation, we find:
acceleration = (75.5 m - 18.0 m) / (0.5 * (1.2 s)^2)
acceleration = 57.5 m / (0.5 * 1.44 s^2)
acceleration = 79.86 m/s^2
Therefore, Angelo must slow down his car at a rate of approximately 79.86 m/s^2 to stop 0.5 m before hitting the stalled car.
To find the rate at which Angelo must slow down his car, we need to consider the distance, speed, and time involved. Let's break down the problem step by step:
1. First, we need to calculate the total distance Angelo needs to cover to stop his car safely. From the problem, we know Angelo wants to stop 0.5 m before hitting the stalled car. Therefore, the total distance he must cover is 75 m + 0.5 m = 75.5 m.
2. We also need to consider Angelo's reaction time. During the reaction time, Angelo is still traveling at his initial speed of 15.0 m/s. So, we need to calculate the distance he travels during this time. The distance covered during the reaction time can be found using the formula: distance = speed * time. Therefore, Angelo covers a distance of 15.0 m/s * 1.2 s = 18.0 m during his reaction time.
3. Now, we can determine the distance Angelo needs to slow down from his initial speed to stop completely. This distance is given by the difference between the total distance and the distance covered during his reaction time: distance to slow down = total distance - distance during reaction time. In this case, distance to slow down = 75.5 m - 18.0 m = 57.5 m.
4. Finally, we can calculate the rate at which Angelo must slow down his car. This rate of deceleration (negative acceleration) can be calculated using the equation: acceleration = (final velocity^2 - initial velocity^2) / (2 * distance). In this case, the final velocity is 0 m/s (since Angelo wants to stop completely), the initial velocity is 15.0 m/s, and the distance is 57.5 m. Plugging these values into the equation, we get: acceleration = (0^2 - 15.0^2) / (2 * 57.5) = -225.0 / 115 = -1.96 m/s^2.
Therefore, Angelo must slow down his car at a rate of approximately 1.96 m/s^2 to stop safely 0.5 m before hitting the stalled car, considering his reaction time of 1.2 s.