Bissoy.com
Doctors Medicines MCQs Q&A

The distance x of a particle moving along a straight line from a fixed point on the line at time t after start is given by t = ax2 + bx + c (a, b, c are positive constants). If v be the velocity of the particle and u(≠0) be the initial velocity of the particle then which one is correct?

The distance x of a particle moving along a straight line from a fixed point on the line at time t after start is given by t = ax2 + bx + c (a, b, c are positive constants). If v be the velocity of the particle and u(≠0) be the initial velocity of the particle then which one is correct? Correct Answer 1/v2 – 1/u2 = 4at

We have, t = ax2 + bx + c  ……….(1) Differentiating both sides of (1) with respect to x we get, dt/dx = d(ax2 + bx + c)/dx = 2ax + b Thus, v = velocity of the particle at time t = dx/dt = 1/(dt/dx) = 1/(2ax + b) = (2ax + b)-1  ……….(2) Initially, when t = 0 and v = u, let x = x0; hence, from (1) we get, ax02 + bx0 + c = 0 Or ax02 + bx0 = -c   ……….(3) And from (2) we get, u = 1/(2ax0 + b) Thus, 1/v2 – 1/u2 = (2ax + b)2 – (2ax0 + b)2 = 4a2x2 + 4abx – 4a2x02 – 4abx0 = 4a2x2 + 4abx – 4a(ax02 – bx0) = 4a2x2 + 4abx – 4a(-c)   = 4a(ax2 + bx + c) Or 1/v2 – 1/u2 = 4at

Related Questions

The distance x of a particle moving along a straight line from a fixed point on the line at time t after start is given by t = ax2 + bx + c (a, b, c are positive constants). If v be the velocity of the particle at the instant, then which one is correct?
How far is point 'R' from Point 'T'? Statement (I): Point 'R' is 5 metres to the north of point 'M'. Point 'U' is 4 metres to the east of point 'R'. Point 'T' is to the west of point 'R' such that points 'U' 'R' and 'T' form a straight line of  metres. Statement (II): Point 'Z' is metres to the south of point 'T'. Point 'U' is  metres to the east of point 'T'. Point 'M' is  metres to the east of point 'Z'. Point 'R' is  metres to the north of point 'M'. Point 'R' lies on the line formed by joining points 'T' and 'U'.
Garima starts walking towards the south from point R. He walks for 5m to reach Point S then turns right and walks for 2m to reach point T from there he turns left and walks for 3m to reach Point Z. He then turns right from point Z and walks for 6m to reach point Y and then turns right and walks for 3m to reach point X. He then turns right from point X and walks for 4m to reach point W and then turns left and walks for 5m to reach point V and then turns left and walks for 4m till point U and stops. What is the shortest distance between point V and point R?
The distance s of a particle moving along a straight line from a fixed-pointO on the line at time t seconds after start is given by x = (t – 1)2(t – 2)2. What will be the distance of the particle from O when its velocity is zero?
Point A is 3m in the north of point B. Point D is 2m in the east of point E. Point F is 9m in the west of point G. Point C is 4m in the west of point B. Point C is 3m in the south of point D. Point F is 5m in the south of point E. Point H is 5m in the north of point G. What is the shortest distance between point H and point A?
Study the following information and answer the questions. Point A is 8m to the west of Point B. Point C is 4m to the south of Point B. Point D is 4m to the east of Point C. Point F is 6m to the north of Point D. Point E is 8m to the west of Point F. Point G is 2m to the south of Point E. If point G is 4m to the north of Point H, then what is the distance between H and D?
Suyash starts walking from Point A and goes 130 m towards southeast. Pradeep starts walking from Point B and goes 130 m towards southwest. After walking these distances, they both meet at Point C. Point D is located in such a way that if points A and B are connected with a straight line, then D would be the midpoint of that line. The distance between Point C and Point D is 120 m. What is the distance between Point A and Point B ?
Point P is 4m in the west of point V. Point R is 6m in the north of point Q. Point T is 5m in the south of point S. Point V is in the middle of point R and Q. Point U is 5m in the east of point R. Point V is in north of point Q. Point S is 5m in the east of point V. What is the area of triangle formed by point P, R and Q?
A particle is moving in a straight line is at a distance x from a fixed point in the straight line at time t seconds, where x = 2t3 – 12t + 11. What is the velocity of the particle at the end of 2 seconds?
A particle is moving in a straight line and its distance s cm from a fixed point in the line after t seconds is given by s = 12t – 15t2 + 4t3. What will be the distance between the two positions of the particle at two times, when the velocity is instantaneously 0?