Question

A cannon is rigidly attached to a carriage,which can move along horizontal rails but is connected to a post by a large spring, initially unstretched and with force constant $k=2.00\times {10}^{4}N/m$, as shown in Figure $P9.70$. The cannon fires a $200-kg$ projectile at a velocity of $125m/s$ directed ${45.0}^o$ above the horizontal.
Find the maximum force the spring exerts on the carriage.

Answer 1

Given,

spring constant $k=2\times {10}^{4}N/m$

velocity of ball is $V=125m/s$

mass of ball is $m=200kg$

angle of projection is ${45}^{\circ }$

Let extension of string is $x$

We know that by energy theorem total change in kinetic energy is equal to total work done by the force.

Here, final kinetic energy is zero therefore,

$\frac{1}{2}m{v}^2=\frac{1}{2}k{x}^{2}m{v}^2=k{x}^2{x}^2=\frac{200\times 125\times 125}{2\times {10}^4}x=12.5m$

maximum force exerted by the spring is

$F=Kx$

$F=2\times {10}^4\times 12.5\times 12.5F=3125000N$