At the scene of an accident, there are skid marks which are 200 feet long showing where a car put on the brakes. The driver swears that he was obeying the speed limit. If we assume that the car braked with a constant deceleration of 16 feet per second squared, determine if the driver was telling the truth.
average speed * time = 200 ft
average speed = (V-0)/2 = V/2
so
t = 200/(V/2) =400/V
V = a t
V = 16 t = 16 (400/V)
V^2 = 16 * 400
V = 4 * 20 = 80 ft/second
80 ft/s ( 3600 s/hr / 5280 ft/mi) = 54.5 miles/hr
To determine if the driver was telling the truth, we can use the equations of motion. The formula to calculate the distance traveled during deceleration is given by:
s = (v^2 - u^2) / (2a)
Where:
- s is the distance traveled (in this case, 200 feet)
- v is the final velocity (0 ft/s, as the car comes to a stop)
- u is the initial velocity (unknown)
- a is the deceleration (16 ft/s^2)
Substituting the known values into the equation, we have:
200 = (0^2 - u^2) / (2 * 16)
Simplifying further:
200 = -u^2 / 32
Multiplying both sides by -32:
-6400 = u^2
Taking the square root of both sides:
u = ± 80
Since velocity is a scalar quantity, we take the positive value:
u = 80 ft/s
The initial velocity (u) is 80 ft/s. Therefore, the driver was not telling the truth because the car was initially traveling at a speed of 80 ft/s, which exceeds the speed limit.
To determine if the driver was telling the truth, we need to calculate the car's initial speed before braking using the given information. We can use the equation of motion:
v^2 = u^2 + 2as
Where:
- v is the final velocity (0 ft/s as the car comes to a stop)
- u is the initial velocity (what we want to find)
- a is the acceleration (deceleration in this case) (-16 ft/s^2)
- s is the displacement (length of the skid marks, 200 ft)
Rearranging the equation, we get:
u^2 = v^2 - 2as
Substituting the values we have:
u^2 = 0 - 2(-16)(200)
u^2 = 6400
Taking the square root of both sides:
u = √6400
u = 80 ft/s
Therefore, the initial speed of the car before braking was 80 ft/s. To determine if the driver was telling the truth, we need to check if this speed was within the speed limit.