Answer
The greatest speed the projectile can have is 90.3 m/s
Work Step by Step
After the wood ball starts moving, the centripetal force will keep the ball moving in a circle as long as the required tension in the wire does not exceed 400 N. We can find the maximum speed of the ball and projectile after the collision.
$\sum F = \frac{mv^2}{R}$
$T - mg = \frac{mv^2}{R}$
$v^2 = \frac{(T - mg)~R}{m}$
$v = \sqrt{\frac{(T - mg)~R}{m}}$
$v = \sqrt{\frac{[400~N - (20~kg+1.0~kg)(9.80~m/s^2)]~(2.0~m)}{20~kg+1.0~kg}}$
$v = 4.3~m/s$
To find the greatest speed of the projectile, we can set the initial momentum of the projectile equal to the final momentum of the ball and projectile when the speed after the collision is 4.3 m/s.
$p_0 = p_f$
$(1.0~kg)~v_0 = (21~kg)(4.3~m/s)$
$v_0 = 90.3~m/s$
The greatest speed the projectile can have is 90.3 m/s.
