A particle is moving in a straight line such that its velocity is increasing at 5 ms⁻¹ per meter. The acceleration of the particle is _____________ ms⁻² at a point where its velocity is 20 ms⁻¹.

Solution

Physics sir · 4 steps

Velocity changing per metre means dv/dx is given, so use a = v dv/dx.

1. Read the rate. The velocity rises by 5 m s⁻¹ for every metre travelled, not every second. That is a rate with respect to distance, so dv/dx = 5 s⁻¹.
2. Link to acceleration. Acceleration is dv/dt. Use the chain rule through position: dv/dt = dv/dx·dx/dt, and dx/dt = v.

   a = v dv/dx

3. Put in numbers. At v = 20 m s⁻¹, a = 20 × 5 = 100 m s⁻².
4. So. The acceleration at that point is 100 m s⁻².

The common slip: 5. Reading 5 per metre as the acceleration itself gives 5, but 5 is dv/dx, and it still has to be multiplied by v = 20.

The idea to keep. Whenever a rate is given per metre instead of per second, it is dv/dx. Multiply by v to get acceleration: a = v dv/dx.

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