A certain cable car is San Francisco can stop in 10 s when traveling at maximum speed. On one occasion, the driver sees a dog a distance d m in front of the car and slams on the brakes instantly. The car reaches the dog 7.78 s later, and the dog jumps off the track just in time. If the car travels 3.58 m beyond the position of the dog before coming to a stop, how far was the car from the dog?
Let's break down the problem step by step:
1. We know that the car takes 10 seconds to stop from maximum speed.
2. The car reaches the dog 7.78 seconds after the brakes are slammed.
3. The car travels an additional 3.58 meters beyond the position of the dog before coming to a stop.
To find the distance between the car and the dog, we need to find two distances - the distance the car covered in 7.78 seconds and the distance it covered in the additional 3.58 meters.
Step 1: Calculate the distance covered in 7.78 seconds:
We can use the formula for distance covered with constant acceleration:
distance = initial velocity * time + (1/2) * acceleration * time^2
Since we're dealing with braking, the acceleration will be negative (deceleration), and the initial velocity is unknown. However, we can assume that the initial velocity is constant, so the formula simplifies to:
distance = initial velocity * time
We don't know the initial velocity, but we know the car is at maximum speed, so we'll use the distance covered during maximum speed (10 seconds) to find the initial velocity:
initial velocity = distance / time
Given that the car traveled an additional 3.58 meters beyond the position of the dog before coming to a stop, we can calculate the initial velocity as:
initial velocity = 3.58 m / 10 s
Step 2: Calculate the distance covered in 7.78 seconds:
distance = initial velocity * time
distance = (3.58 m / 10 s) * 7.78 s
Now we have the distance covered in 7.78 seconds.
Step 3: Calculate the total distance between the car and the dog:
total distance = distance covered in 7.78 seconds + additional distance covered
total distance = (3.58 m / 10 s) * 7.78 s + 3.58 m
Now we can calculate the total distance.
Simplifying the equation, we have:
total distance = 2.7776 m + 3.58 m
total distance = 6.3576 m
Therefore, the car was approximately 6.36 meters from the dog.
To solve this problem, we can use the equations of motion, which relate the distance traveled (d), initial velocity (u), final velocity (v), acceleration (a), and time (t). In this case, the initial velocity of the car is the maximum speed it travels at. Let's break down the given information:
Initial velocity of the car (u) = maximum speed of the car
Final velocity of the car (v) = 0 m/s (since the car comes to a stop)
Time taken to come to a stop (t) = 10 s
Time taken to reach the dog (t1) = 7.78 s
Distance traveled beyond the dog (d1) = 3.58 m
Now, we'll use the equations of motion to find the acceleration (a).
First equation: v = u + at
Since v = 0 (car comes to a stop), we can rearrange the equation:
0 = u + a * 10 (equation 1)
Now, let's use the second equation of motion to find the distance traveled (d1) in the time it takes to reach the dog (t1).
Second equation: d1 = ut1 + (1/2) * a * t1^2 (equation 2)
We also know that the car travels an additional distance (d2) beyond the dog. So, the total distance traveled (d) is the sum of d1 and d2.
Total distance traveled (d) = d1 + d2 (equation 3)
Now, let's solve the equations step by step.
Using equation 1: 0 = u + a * 10
Since we know u = maximum speed, we can substitute it into the equation:
0 = maximum speed + a * 10
Using equation 2: d1 = ut1 + (1/2) * a * t1^2 (equation 2)
Since we again know u = maximum speed, we can substitute it into the equation:
d1 = maximum speed * 7.78 + (1/2) * a * (7.78)^2
Now, we'll solve for a in equation 1 and substitute it in equation 2 to find the value of d1.
0 = maximum speed + a * 10
a = -maximum speed / 10 (substituting)
d1 = maximum speed * 7.78 + (1/2) * (-maximum speed / 10) * (7.78)^2
Now, let's rearrange equation 3 to solve for d:
d = d1 + d2
We are given that d2 = 3.58 m.
So,
d = [maximum speed * 7.78 + (1/2) * (-maximum speed / 10) * (7.78)^2] + 3.58
By substituting the appropriate values, we can calculate the distance (d) between the car and the dog.