To stop a car, first you require a certain reaction time to begin braking. Then the car slows down at a constant rate. Suppose that the total distance moved by your car during these two phases is 56.7 m when its initial speed is 85.0 km/h, and 24.4 m when the initial speed is 52.8 km/h.
(a) What is your reaction time?
(b) What is the magnitude of the deceleration?
85.0 km/h = 23.61 m/s
52.8 km/h = 14.67 m/s
Total distance = reaction time distance + braking distance
= V*t + V^2/(2a)
Write two equations in two unknowns and solve for the two unknowns, a and t.
56.7 = 23.61 t - (23.61)^2/(2a)
24.4 = 14.67 t - (14.67)^2/(2a)
To solve this problem, we need to use the equations of motion.
(a) To find the reaction time, we need to determine how long it takes for the car to start slowing down after the brakes are applied. We can use the formula:
d = v_i * t + (1/2) * a * t^2
where:
d is the distance moved during the reaction time,
v_i is the initial velocity,
t is the reaction time, and
a is the acceleration.
Given that the car moves 56.7 m during the reaction time and the initial velocity is 85.0 km/h, we need to convert the velocity to m/s:
v_i = 85.0 km/h * (1000 m / 1 km) * (1 h / 3600 s) = 23.6 m/s
Plugging in the values, we have:
56.7 m = 23.6 m/s * t + (1/2) * a * t^2
We also know that during the reaction time, the car starts from rest:
t = 0
Thus, the equation becomes:
56.7 m = 0 + 0
So, we don't have any additional information to determine the reaction time.
(b) To find the magnitude of the deceleration, we can use the formula:
d = (v_f^2 - v_i^2) / (2a)
where:
d is the distance moved during the deceleration,
v_f is the final velocity, and
a is the deceleration.
Given that the car moves 24.4 m during the deceleration and the initial velocity is 52.8 km/h, we need to convert the velocity to m/s:
v_i = 52.8 km/h * (1000 m / 1 km) * (1 h / 3600 s) = 14.7 m/s
Plugging in the values, we have:
24.4 m = (0 - 14.7 m/s)^2 / (2a)
Simplifying the equation:
24.4 m = 214.6 m^2/s^2 / (2a)
48.8 m * a = 214.6 m^2/s^2
a = (214.6 m^2/s^2) / 48.8 m
a = 4.40 m/s^2
Therefore, the magnitude of the deceleration is 4.40 m/s^2.
To solve this problem, we can use the kinematic equations of motion. Let's break it down step by step:
(a) To find the reaction time, we need to determine the distance traveled during the reaction time before braking starts. This distance is the difference between the total distance moved by the car and the distance covered during the constant deceleration phase.
For the first case, when the initial speed is 85.0 km/h:
1. Convert the initial speed from km/h to m/s.
- 85.0 km/h * (1000 m/1 km) * (1 h/3600 s) = 23.6 m/s
2. The distance covered during the constant deceleration phase is given as 56.7 m.
3. Subtract the distance covered during the deceleration phase from the total distance to find the distance covered during the reaction time phase.
- Reaction time distance = Total distance - Deceleration distance
= 56.7 m - 23.6 m
= 33.1 m
Now we can calculate the reaction time using the equation for motion during constant velocity:
Distance = Initial velocity * Time
33.1 m = 23.6 m/s * Time
Solve for Time:
Time = 33.1 m / 23.6 m/s
= 1.40 s
Therefore, the reaction time is approximately 1.40 seconds.
(b) To find the magnitude of the deceleration, we can use the equation for motion during constant deceleration:
Distance = Initial velocity * Time + (1/2) * Acceleration * Time^2
For the second case, when the initial speed is 52.8 km/h:
1. Convert the initial speed from km/h to m/s.
- 52.8 km/h * (1000 m/1 km) * (1 h/3600 s) = 14.7 m/s
2. The distance covered during the reaction time phase is given as 24.4 m.
3. Use the distance equation to solve for the deceleration.
24.4 m = 14.7 m/s * Time + (1/2) * Acceleration * Time^2
Substitute the known values and rearrange the equation:
Acceleration * Time^2 + (14.7 m/s) * Time - 24.4 m = 0
This quadratic equation can be solved using various methods, such as factoring, completing the square, or using the quadratic formula. Once solved, you can find the value of acceleration.
Please note that without additional information, we cannot provide the exact value for the magnitude of the deceleration in this example.