A dart is loaded into a spring-loaded toy dart gun by pushing the spring in by a distance d. For the next loading, the spring is compressed a distance 8d. How much work is required to load the second dart compared to that required to load the first?

the work required for the loading of second dart is 64 times greater as work required for loading the first dart.

Explanation:

k = spring constant of the spring loaded toy dart gun

x₁ = compression of spring to load the first dart = d

x₂ = compression of spring to load the second dart = 8 d

E₁ = Work required to load the first dart

E₂ = Work required to load the second dart

Work required to load the first dart is given as

E₁ = (0.5) k x₁² = (0.5) k d²

Work required to load the second dart is given as

E₂ = (0.5) k x₂² = (0.5) k (8d) ² = (64) (0.5) k d²

E₂ = 64 E₁

So the work required for the loading of second dart is 64 times greater as work required for loading the first dart