If the velocity of a body related to displacement x is given by υ = √(5000 + 24x) m/s, then the acceleration of the body is .................... m/s².

Solution

Physics sir · 4 steps

Square the velocity first, then a = 1/2d(v²)/dx gives the acceleration in one line.

1. Remove the root. Square both sides: v² = 5000 + 24x. Now v² is a simple straight-line function of x.
2. Acceleration from v(x). When velocity is given in terms of displacement, use a = vdv/dx. Since d(v²)/dx = 2vdv/dx, this is the same as half the derivative of v².

   a = vdv/dx = 1/2d(v²)/dx = 1/2 × 24 = 12 m s⁻²

3. Check with kinematics. Compare with v² = u² + 2ax: here u² = 5000 and 2a = 24. The acceleration is constant and equal to 12 m s⁻².
4. So. The acceleration of the body is 12 m s⁻².

The common slip: 24. Taking the coefficient of x in v² as the acceleration gives 24. That coefficient is 2a, so halve it to get 12.

The idea to keep. When v is given as a function of x, use a = vdv/dx = 1/2d(v²)/dx. If v² is linear in x, the acceleration is constant.

Answer from the NTA final answer key. Solution by PrepWiser.