Police find skid marks 60 m long on a road showing where a car made an emergency stop. Assuming that the deceleration was -10m/s2, how fast was the car going?
Well, it seems like this car had a serious case of "braking bad"! Let's calculate its initial speed, shall we?
We know that the distance covered during the emergency stop is 60 meters, and the deceleration is -10 m/s^2. To find the initial speed, we can use the following equation:
v^2 = u^2 + 2as
Where:
v = final velocity (0 m/s since the car stopped)
u = initial velocity (what we want to find)
a = acceleration (-10 m/s^2)
s = distance covered (60 meters)
Rearranging the equation, we have:
u^2 = -2as
Plugging in the values, we get:
u^2 = -2(-10)(60)
u^2 = 1200
u ≈ √1200
u ≈ 34.64 m/s
So, it seems like the car was initially going at approximately 34.64 m/s. Guess it had to hit the brakes due to some "speedbump" in the road!
To find the speed of the car, we can use the equation of motion:
v^2 = u^2 + 2as
Where:
v = final velocity (0 m/s because the car comes to a stop)
u = initial velocity (the speed we want to find)
a = acceleration (-10 m/s^2)
s = distance (60 m)
Rearranging the equation, we get:
u^2 = v^2 - 2as
Plugging in the values:
u^2 = 0^2 - 2(-10)(60)
Simplifying:
u^2 = 1200
Taking the square root of both sides:
u = ±√1200
Since speed cannot be negative, we take the positive root:
u ≈ √1200
Calculating:
u ≈ 34.64 m/s
Therefore, the car was going approximately 34.64 m/s.
To find the speed of the car, we can use the equation of motion that relates distance, initial velocity, acceleration, and time.
The equation is:
distance = (initial velocity × time) + (0.5 × acceleration × time²)
In this case, the car came to a stop, so the final velocity is 0 m/s. Therefore, the equation becomes:
distance = initial velocity × time + 0.5 × acceleration × time²
We know that the skid marks are 60 m long and the deceleration is -10 m/s². We need to find the initial velocity, which is the speed of the car.
Let's solve the equation for the initial velocity:
60 m = initial velocity × time + 0.5 × (-10 m/s²) × time²
Simplifying the equation, we have:
60 m = initial velocity × time - 5 m/s² × time²
Now, we need another equation to relate time and initial velocity. We can use the fact that the final velocity is zero:
final velocity = initial velocity + (acceleration × time)
With the final velocity being 0 m/s and the acceleration being -10 m/s², the equation becomes:
0 m/s = initial velocity + (-10 m/s²) × time
We can rearrange the equation to solve for time:
initial velocity = 10 m/s² × time
Now we have a system of equations, combining both equations:
60 m = (10 m/s² × time) × time - 5 m/s² × time²
Simplifying further:
60 m = 10 m/s² × time² - 5 m/s² × time²
Combining like terms:
60 m = 5 m/s² × time²
Dividing both sides of the equation by 5 m/s²:
12 = time²
Taking the square root of both sides:
√12 ≈ 3.46 seconds
Now we know the time it took for the car to stop is approximately 3.46 seconds.
Let's substitute this value back into one of the equations to find the initial velocity:
initial velocity = 10 m/s² × 3.46 s
Calculating:
initial velocity ≈ 34.6 m/s
Therefore, the car's initial speed, or the speed before the emergency stop, was approximately 34.6 m/s.