Question

A catapult with a basket of mass 50 kg launches a 200 kg rock by swinging around from a horizontal to a vertical position with an angular velocity of 2.0 rad/s. Assuming the rest of the catapult is massless and the catapult arm is 10 m long, what is the velocity of the rock as it leaves the catapult?
A. $10m/s$
B. $20m/s$
C. $25m/s$
D. $50m/s$
E. $100m/s$

Answer 1

This question can be answered by using newton’s second law. According to Newton's second law for a particular system rate of change of momentum will be equal to the external force acting on the system. Here we consider catapult and rock as a single system at the time rock leaves.

Formula used:
${F_{ext}} = \dfrac{{d{p_{system}}}}{{dt}}$
$v = l \times w$

Complete step by step answer:
First let us consider catapult and rock as a single isolated system.
Now at the instant rock leaves catapult only dominant force acting on catapult is normal force created by rock and only dominant force acting on rock is normal force created by catapult. Both will be in mutually opposite directions and in the same magnitude.
Since we considered rock and catapult as a single system these two forces will become internal forces and cancel out each other and there are no external forces(${F_{ext}} = 0$).
According to newton’s second law ${F_{ext}} = \dfrac{{d{p_{system}}}}{{dt}} = 0$
That change in momentum of the system will be zero.
$d{p_{sys}} = 0$
Initial momentum of system = final momentum of system
Initial condition means just before throwing rock and final condition means just after throwing it.
Initial linear velocity(v) of a system is
$v = l \times w$
Where ‘l’ is the length of the arm and ‘w’ is the angular velocity
l = 10m, w = 2 rad/s
$v = 10 \times 2 = 20m/s$
From conservation of momentum
$(m + M)v = M{v^1}$
M = mass of catapult
m = mass of rock
v = initial system velocity
${v^1}$ = velocity of rock just after leaving
M = 200kg, m = 50kg, v = 20m/s
So
${v^1} = (\dfrac{{50 + 200}}{{200}})(20) = 25m/s$

So, the correct answer is “Option C”.

Note: According to Newton’s second law external force will be zero only if we consider rock and catapult as a single system. If we consider rock and catapult as different systems then normal force acted on catapult due to rock will be external force for catapult and force acted on rock due to catapult will be external force for rock. At the time of leaving only rock that has velocity catapult will be at rest.