The driver of a 2.0 × 103 kg red car traveling on the highway at 45m/s slams on his brakes to avoid striking a second yellow car in front of him, which had come to rest because of blocking ahead as shown in above Fig. After the brakes are applied, a constant friction force of 7.5 × 103 N acts on the car. Ignore air resistance.
(a) Determine the least distance should the brakes be applied to avoid a collision with the other vehicle?
(b) If the distance between the vehicles is initially only 40.0 m, at what speed would the collision occur?
(c) Write your conclusive observations on the result obtained from this numerical. i.e. the importance of Physics in daily life.
Newton's second law says:
7.5*10^3 = 2*10^3 * a
a = -7.5 /2 (negative) if distance moved is positive
v = Vi + a t = 45 - 7.5 t/2
when v = 0
7.5 t = 90
t = 12 seconds to stop
x = Vi t + (1/2) a t^2
x = 45*12 - 7.5/4 (144) = 540 - 270 = 270 meters to stop
now if x = 40
then
40 = 45 t - 7.5/4 t^2
1.875 t^2 - 45 t + 40 = 0
solve quadratic for t
then
v = 45 - (7.5/2) t
(c) crash !
To answer these questions, we need to use principles of physics such as Newton's laws of motion and equations of motion. Let's break down each part:
(a) To determine the least distance required to avoid a collision, we need to find the stopping distance of the red car. We can use the equations of motion for an object undergoing uniform acceleration.
The equation we will use is:
v_f^2 = v_i^2 + 2ad
where v_f is the final velocity (0 m/s, as the car is coming to a stop), v_i is the initial velocity (45 m/s), a is the acceleration (equal to the friction force divided by the mass of the car), and d is the stopping distance.
Rearranging the equation to solve for d, we have:
d = (v_f^2 - v_i^2) / (2a)
we substitute the values, we get:
d = (0^2 - 45^2) / (2 * (7.5 * 10^3 N) / (2 * 10^3 kg))
Calculating this, we find that the least distance the brakes should be applied to avoid a collision is approximately 135 meters.
(b) If the distance between the vehicles is initially only 40.0 m, we need to determine if a collision will occur. We can find this by comparing the stopping distance (calculated in part a) to the initial distance between the two cars. If the stopping distance is greater than the initial distance, a collision will occur.
In this case, since the stopping distance is 135 meters and the initial distance is 40 meters, a collision will occur.
(c) The result obtained from this numerical problem demonstrates the importance of physics in daily life, especially when it comes to understanding and ensuring safety on the roads. By using principles of physics, we can determine the minimum stopping distance required to avoid a collision. This knowledge is crucial for drivers to operate their vehicles safely and make informed decisions on the road. Physics helps us understand the motion of objects and the forces involved, allowing us to predict and prevent potential accidents.