A particle moves such that its position vector r⃗ ( t ) = cos ω tî + sin ω tĵ where ω is a constant and t is time. Then which of the following statements is true for the velocity v⃗ ( t ) and acceleration a⃗ ( t ) of the particle :

Solution

Physics sir · 4 steps

Differentiate twice and compare with r⃗: the acceleration comes out as -ω² r⃗.

1. Find velocity. Differentiate r⃗ once: v⃗ = -ω sin ω t î + ω cos ω t ĵ. Then v⃗ · r⃗ = -ω sin ω t cos ω t + ω sin ω t cos ω t = 0, so v⃗ is perpendicular to r⃗.
2. Find acceleration. Differentiate once more: a⃗ = -ω² cos ω t î - ω² sin ω t ĵ. Each component is -ω² times the matching component of r⃗.

   a⃗ = -ω² r⃗

3. Read the sign. r⃗ points from the origin to the particle. The minus sign means a⃗ points the opposite way, from the particle back towards the origin. This is centripetal acceleration for motion on a circle of radius 1.
4. So. v⃗ is perpendicular to r⃗ and a⃗ is directed towards the origin, which is option (c).

The idea to keep. In uniform circular motion, velocity is along the tangent, at right angles to r⃗. Acceleration is -ω² r⃗: always along the radius, pointing towards the centre.

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