cars A and B are d=60m apart and traveling respectively with constant speeds v1=32km/h and v2=24km/h on an ice covered road.knowing that 45s after driver A apples his breaks to avoid overtaking car Bthe two cars collide,determine a)the uniform deceleration of car A b)the relative velocity of car A with respect to car B when they collide
To solve this problem, we need to analyze the motion of the two cars separately and then determine the conditions for the collision.
Let's start with analyzing the motion of Car A:
Step 1: Convert the speeds to m/s.
v1 = 32 km/h = 32 * (1000/3600) m/s = 8.89 m/s
Step 2: Calculate the time when Car A applies its brakes.
Given that the collision occurs 45 seconds after Car A applies its brakes, let's denote this time as t. So, t = 45s
Step 3: Calculate the initial velocity of Car A.
The initial velocity of Car A can be calculated using the equation of motion: v = u + at, where v is the final velocity, u is the initial velocity, a is the deceleration, and t is the time.
Since Car A comes to a stop, v = 0 m/s. Therefore, the equation becomes: 0 = u + at.
Rearranging the equation, we get: u = -at
Step 4: Calculate the deceleration (a) of Car A.
Using the equation: v = u + at, we can substitute the known values:
0 = -at + at
0 = (-a)(t - t)
0 = (-a)(0)
Since the acceleration (-a) cannot be zero, it means that the deceleration (a) is also zero.
Therefore, the uniform deceleration of Car A is zero (a = 0).
Now let's move on to analyzing the motion of Car B:
Step 1: Convert the speed to m/s.
v2 = 24 km/h = 24 * (1000/3600) m/s = 6.67 m/s
Step 2: Calculate the distance traveled by Car B during the time t.
Car B travels a distance equal to its speed multiplied by the time t, which is given by:
distance = speed * time
distance = 6.67 m/s * 45 s = 300.15 m
Step 3: Determine the relative velocity of Car A with respect to Car B when they collide.
The relative velocity is simply the difference between the velocities of the two cars, as they are moving in the same direction:
relative velocity = velocity of Car A - velocity of Car B
relative velocity = v1 - v2
relative velocity = 8.89 m/s - 6.67 m/s
relative velocity ≈ 2.22 m/s
Therefore, the answers are:
a) The uniform deceleration of Car A is zero (a = 0).
b) The relative velocity of Car A with respect to Car B when they collide is approximately 2.22 m/s.
To find the answers, we need to use the kinematic equations for motion. Let's break down the problem step by step.
Step 1: Convert the given speeds from km/h to m/s:
v1 = 32 km/h = 32 * (1000/3600) m/s ≈ 8.89 m/s
v2 = 24 km/h = 24 * (1000/3600) m/s ≈ 6.67 m/s
Step 2: Determine the time it takes for the cars to collide:
Car A applies the brakes 45 seconds before the collision, so the time taken by car A is 45 seconds.
Step 3: Calculate the acceleration of car A:
Let a be the uniform deceleration of car A.
Using the kinematic equation:
d = v0 * t + (1/2) * a * t^2
where d is the distance, v0 is the initial velocity, t is the time, and a is the acceleration.
For car A:
d = 60 m (distance between the cars)
v0 = 8.89 m/s (initial velocity of car A)
t = 45 s (time taken by car A)
We can rearrange the equation to solve for a:
a = (2*(d - v0*t))/(t^2)
a = (2*(60 - 8.89*45))/(45^2)
a ≈ -0.3622 m/s^2 (approximately)
Therefore, the uniform deceleration of car A is approximately -0.3622 m/s^2. The negative sign indicates that it is decelerating.
Step 4: Calculate the relative velocity of car A with respect to car B when they collide:
The relative velocity (v_rel) of car A with respect to car B is the difference between their velocities:
v_rel = v1 - v2
v_rel = 8.89 m/s - 6.67 m/s
v_rel ≈ 2.22 m/s
Therefore, the relative velocity of car A with respect to car B when they collide is approximately 2.22 m/s.