Question:

A car starts from rest and accelerates uniformly at a rate of 2 m/s^2 for a distance of 50 meters. After reaching this distance, the car decelerates uniformly at a rate of -1 m/s^2 until it comes to a stop. Calculate the displacement of the car during this entire motion.

(Assume the positive direction is taken as forward motion)

Answer:

Given: Initial velocity, u = 0 m/s Acceleration, a1 = 2 m/s^2 (for the first 50 meters) Deceleration, a2 = -1 m/s^2 (until the car stops)

We can calculate the displacement of the car in two different segments:

Segment 1: Constant acceleration until distance = 50 meters

Using the second equation of motion:

v1^2 = u^2 + 2a1s1

Where: v1 = Final velocity in segment 1 u = Initial velocity in segment 1 a1 = Acceleration in segment 1 s1 = Distance traveled in segment 1

Since the car starts from rest (u = 0), the equation simplifies to:

v1^2 = 2a1s1

Final velocity in segment 1 can be calculated using the first equation of motion:

v1 = u + a1t1

Where: t1 = Time taken in segment 1

Since the car starts from rest, initial velocity (u) in segment 1 is zero:

v1 = a1t1

Substituting this value in the first equation:

(a1t1)^2 = 2a1s1

Simplifying the equation:

a1^2t1^2 = 2a1s1

Dividing both sides of the equation by a1:

t1^2 = 2s1/a1

Simplifying further:

t1 = √(2s1/a1)

Segment 2: Constant deceleration until the car comes to a stop (distance = s2)

Using the third equation of motion:

v2^2 = u^2 + 2a2s2

Where: v2 = Final velocity in segment 2 u = Initial velocity in segment 2 a2 = Deceleration in segment 2 s2 = Distance traveled in segment 2

Since the final velocity in segment 2 is zero (car comes to a stop), the equation becomes:

0 = u^2 + 2a2s2

As the car starts from rest (u = 0), the equation simplifies to:

0 = 2a2s2

Dividing both sides of the equation by 2a2:

s2 = 0

The displacement of the car during the entire motion is the sum of the displacements in segments 1 and 2:

Displacement = s1 + s2

Since s2 = 0, we get:

Displacement = s1

Therefore, the displacement of the car during this entire motion is 50 meters.