A car is moving at 90 km/h when the driver notices an animal that is about to cross the road 100 m away and he decides to apply the brakes. The reaction time of the driver is 0.3 s.
Calculate the acceleration of the car when it stopped and the time taken from the time the driver noticed the animal to the time the car stopped
Vo = 90km/h = 90000m/3600s. = 25 m/s.
V^2 = Vo^2 + 2a*d. Solve for a.
a = (V^2-Vo^2)/2d = (0-(25^2)/200 = -3.125 m/s^2.
t = 0.3 + (-Vo/a) = 0.3 + (-25/-3.125) =
8.3 s.
Well, if the driver is like me, it might take a little longer to react because I would probably shout, "Watch out, Mr. Animal!" before hitting the brakes. But let's assume the driver's reaction time is 0.3 seconds.
First, let's convert the speed of the car from kilometers per hour to meters per second. There are 1000 meters in a kilometer and 3600 seconds in an hour, so we can calculate:
90 km/h = (90 * 1000) / 3600 m/s = 25 m/s
The initial velocity of the car is 25 m/s, and it needs to stop in order to avoid the animal.
Now, let's calculate the time it takes for the car to stop. Since the car is decelerating, we can use the formula:
v = u + at
Where:
v = final velocity (0 m/s)
u = initial velocity (25 m/s)
a = acceleration (unknown)
t = time taken
Plug in the values:
0 = 25 + a * t
Rearrange the equation:
a * t = -25
Now, we need to calculate the distance traveled during the reaction time of 0.3 seconds.
d = ut + 0.5at^2
Where:
d = distance traveled (unknown)
u = initial velocity (25 m/s)
t = time taken (0.3 seconds)
a = acceleration (unknown)
We can substitute some values and solve for d:
d = 25 * 0.3 + 0.5 * a * (0.3)^2
d = 7.5 + 0.045a
Now, let's calculate the distance remaining to stop after the reaction time:
d = 100 m - 7.5 m = 92.5 m
Since we know distance, initial velocity, and final velocity, we can use the equation:
v^2 = u^2 + 2ad
Plug in the values:
0^2 = 25^2 + 2 * a * 92.5
0 = 625 + 185a
185a = -625
a = -625 / 185
a ≈ -3.38 m/s^2
So, the acceleration of the car when it stopped is approximately -3.38 m/s^2 (negative because it's decelerating).
Now, to calculate the time taken from the time the driver noticed the animal to the time the car stopped, we can use the equation:
v = u + at
0 = 25 + (-3.38) * t
-25 = -3.38t
t ≈ 7.40 seconds
So, it took approximately 7.40 seconds from the time the driver noticed the animal to the time the car stopped. Just enough time to think, "Oh no, I hope the animal is okay!" before coming to a halt!
To calculate the acceleration of the car when it stopped, we need to find the deceleration. Deceleration is the negative acceleration that opposes the motion of the car.
First, let's convert the speed from km/h to m/s:
90 km/h = 90,000 m/ 3600 s = 25 m/s.
Next, we will calculate the time taken from the time the driver noticed the animal to the time the car stopped. The reaction time is given as 0.3 s.
Now, we can use the motion equation:
v^2 = u^2 + 2as,
where:
- v is the final velocity, which is 0 m/s (since the car stops),
- u is the initial velocity, which is 25 m/s,
- a is the deceleration of the car,
- s is the distance traveled, which is given as 100 m.
Plugging in the values:
0^2 = (25)^2 + 2a(100),
Simplifying:
0 = 625 + 200a,
Rearranging:
200a = -625,
a = -625 / 200,
a = -3.125 m/s^2.
Therefore, the acceleration of the car when it stopped is -3.125 m/s^2.
To solve this problem, we'll need to break it down into two parts: calculating the time it takes for the driver to react and then calculating the acceleration of the car.
1. Calculating the reaction time:
The reaction time is the time it takes for the driver to perceive the danger and physically react. In this case, the reaction time of the driver is given as 0.3 seconds.
2. Calculating the distance traveled during the reaction time:
To calculate the distance traveled during the reaction time, we need to use the formula:
Distance = Initial Velocity x Time + 0.5 x Acceleration x Time²
In this case:
Initial Velocity (v₀) = 90 km/h = 90,000 m/3600 s = 25 m/s
Time (t) = reaction time = 0.3 s
Acceleration (a) = unknown
Substituting the values into the formula, we get:
Distance = 25 m/s x 0.3 s + 0.5 x a x (0.3 s)²
Distance = 7.5 m + 0.045 a
3. Calculating the stopping distance:
The stopping distance is the distance required by the car to come to a complete stop after applying the brakes. In this case, the stopping distance is equal to the initial distance between the car and the animal (100 m) minus the distance traveled during the reaction time.
Stopping Distance = Initial Distance - Distance Traveled During Reaction Time
Stopping Distance = 100 m - (7.5 m + 0.045 a)
4. Calculating the acceleration of the car:
The acceleration of the car can be calculated using the formula:
Final Velocity (v)² = Initial Velocity (v₀)² + 2 x Acceleration (a) x Distance (d)
In this case:
Final Velocity (v) = 0 m/s (since the car comes to a stop)
Initial Velocity (v₀) = 25 m/s
Distance (d) = Stopping Distance
Substituting the values into the formula, we get:
0 m/s = (25 m/s)² + 2 x a x Stopping Distance
0 = 625 m²/s² + 2 x a x (100 m - (7.5 m + 0.045 a))
Simplifying the equation, we get:
0 = 625 + 200a - 15a - 0.09a²
Now, we can solve this quadratic equation to find the value of 'a'.