The distance x covered by a particle in one-dimensional motion varies with time as x²=at²+2bt+c. If its acceleration depends on x as x⁻ⁿ, where n is an integer, find n.
Physics · Kinematics Part-1 (one dimension) · Acceleration and Avg acceleration
JEE Mains

Question

The distance x covered by a particle in one-dimensional motion varies with time as x²=at²+2bt+c. If its acceleration depends on x as x⁻ⁿ, where n is an integer, find n.

G1 Physics · Accepted Answer

From x²=at²+2bt+c, differentiating gives xv=at+b. Squaring: x²v²=(at+b)²=a(at²+2bt+c)+(b²-ac)=ax²+K, so v²=a+K/x². Therefore acceleration =v dv/dx=(1/2)d(v²)/dx=-K/x³ ∝x⁻³. Thus n=3.