A motor car moving at a speed of 72 km/h can not come to a stop in less than 3 s while for a truck this time interval is 5.0s . On a highway the car is behind the truck both moving at 72 km/h . The truck gives a signal that it is going to stop at emergency. At what distance the car should be from the truck so that it does not bump onto the truck . Human response time is 0.5s
deceleration rate of the truck is
at = 20 m/s / 5s = 4 m/s²
deceleration rate of the car is
at = 20 m/s / 3s = 6⅔ m/s²
As the car can stop much quicker than the truck, then to avoid collision the car and truck velocities will be identical and non zero while the distance between them is near zero with a braking time (t) for the truck and (t - 0.5) for the car
velocity of the truck
vt = 20 - 4t
velocity of the car
vc = 20 - 6⅔(t - 0.5)
vc = 20 - 6⅔t + 3⅓
vc = 23⅓ - 6⅔t
to find the time when both of these are true, set them equal
23⅓ - 6⅔t = 20 - 4t
3⅓ = 2⅔t
t = 1.25 s
In 1.25 seconds the car will have traveled a distance before braking occurs and a distance after braking starts
dc = v₀t₀ + v₀t₁ + ½at₁²
dc = 20(0.5) + 20(0.75) - ½(6⅔)(0.75)²
dc = 23.125 m
in 1.25 seconds, the truck will have traveled
dt = 20(1.25) - ½(4)(1.25)²
dt = 21.875 m
so the minimum follow distance is
23.125 - 21.875 = 1.25 m <=== ANSWER
At 1.25 seconds, both vehicles will be traveling 15 m/s
20 - 4(1.25) = 15
20 - 6⅔(1.25 - 0.5) = 15
For times greater than 1.25 seconds, the car is traveling slower than the truck and the gap between the car and truck increases.
To find the distance at which the car should be from the truck in order to avoid bumping into it, we need to consider the reaction time of the human driver, as well as the time it takes for the car to come to a stop.
First, convert the speed of both the car and the truck from km/h to m/s. Since 1 km/h is equal to 0.27778 m/s, the speed of both the car and the truck is:
Speed = 72 km/h * 0.27778 m/s = 20 m/s
Next, let's calculate the distance covered by the car and the truck during the reaction time of the driver. Given that the reaction time is 0.5 seconds, the distance covered by both the car and the truck during this time is:
Distance = Speed * Time = 20 m/s * 0.5 s = 10 meters
Now, let's calculate the distance covered by the car during the time it takes to come to a complete stop. Given that the stopping time for the car is 3 seconds, we can use the equation for uniformly accelerated motion:
Distance = (Initial Velocity * Time) + (0.5 * Acceleration * Time^2)
Since the car is coming to a stop, the initial velocity is 20 m/s, the time is 3 seconds, and the acceleration can be calculated using the formula Acceleration = (Final Velocity - Initial Velocity) / Time:
Acceleration = (0 - 20 m/s) / 3 s = -6.67 m/s^2 (negative sign indicates deceleration)
Now we can substitute these values back into the equation to find the distance covered by the car during the stopping time:
Distance = (20 m/s * 3 s) + (0.5 * -6.67 m/s^2 * (3 s)^2) = 30 meters - 30 meters = 0 meters
From the calculation above, we can see that the car will come to a stop exactly at the point where it starts braking.
Therefore, the car should be at least 10 meters behind the truck to ensure it does not bump into it, considering the reaction time of the driver and the time it takes for the car to come to a stop.