A girl is skipping stones across a lake. One of the stones...

A girl is skipping stones across a lake. One of the stones accidentally ricochets off a toy boat that is initially at rest in the water (see the drawing). The 0.072 -kg stone strikes the boat at a velocity of $13 \mathrm{m} / \mathrm{s}, 15^{\circ}$ below due east, and ricochets off at a velocity of 11 $\mathrm{m} / \mathrm{s}$ , $12^{\circ}$ above due east. After being struck by the stone, the boat's velocity is $2.1 \mathrm{m} / \mathrm{s},$ due east. What is the mass of the boat? Assume the water offers no resistance to the boat's motion.

A girl is skipping stones across a lake. One of the stones accidentally ricochets off a toy boat that is initially at rest in the water (see the drawing). The 0.072 -kg stone strikes the boat at a velocity of
$13 \mathrm{m} / \mathrm{s}, 15^{\circ}$ below due east, and ricochets off at a velocity of 11 $\mathrm{m} / \mathrm{s}$ , $12^{\circ}$ above due east. After being struck by the stone, the boat's velocity is $2.1 \mathrm{m} / \mathrm{s},$ due east. What is the mass of the boat? Assume the water offers no resistance to the boat's motion.

Key Concepts

Conservation of Linear Momentum

This principle states that the total momentum of a closed system remains constant if no external forces act on it. In collision problems, such as the one described, the momentum before the collision must equal the momentum after the collision in each independent direction.

Vector Decomposition

In two-dimensional collisions, velocities and momenta are vector quantities. Decomposing these vectors into their respective horizontal and vertical components is essential to apply the conservation laws separately for each axis, allowing the problem to be solved using scalar equations for each direction.

Collision Dynamics

This concept involves analyzing how objects interact during a collision, including understanding how momentum is transferred between objects. In scenarios with minimal external forces, such as frictionless environments, momentum conservation enables the prediction of post-collision speeds and directions.

Transcript

00:01 Okay, and this problem with stone is being thrown into the water or she's skipping stones and it hits a boat and ricochets off the boat in a different direction.

00:14 Let me draw that a little more straight.

00:18 Okay, so the question is asking for the mass of the boat because we're given how fast the boat moves after being hit.

00:25 So this is the conservation of momentum problem where the initial momentum is equal to the final momentum.

00:32 Momentum is a vector, so we have to consider direction.

00:36 Initially, we have, i'll just write out in symbols first, p -o of the stone plus p initial of the boat, it's going to equal p -final of the stone plus p -final of the boat.

00:52 P initials of the boat, and these are all vectors, because they have direction.

00:57 P initial of the boat is zero, because the boat is initially at rest, and so p initial of the stone is going to equal p final of the stone plus p final of the boat.

01:07 So these are vectors and we have to consider that.

01:10 What we can do is break each vector into its components.

01:14 We are given the angle from the horizontal that each vector velocity vector is.

01:21 In the first case, it's 15 degrees.

01:24 And in the second it's 12.

01:27 So we can draw vector components.

01:31 And because the water is not going to allow the boat to move down because of the force, the buoyant force, we can just analyze the horizontal motion of the system.

01:43 So if this is our initial velocity of the stone, and this is our final velocity of the stone, what we can look at is the x component.

01:55 So the component here and the component here, and we can ignore the downward and upward momentum change of the stone, because there would be an impulse applied from the water.

02:08 So looking at that, we know that after the collision, the boat will have a final velocity.

02:17 So therefore, we'll have a final momentum.

02:19 So looking at this mathematically or using trig, if we can find the initial momentum of the stone and the final, momentum of the stone, we could use trache to find those horizontal components, which we're looking for.

02:31 So i'm going to redraw the picture, so it's a little bit easier to see.

02:37 My initial momentum of the stone is going to be the mass of the stone times the velocity and initial velocity of the stone.

02:48 So we have, i guess i'll do this over here.

02:51 Mass of the stone is 0 .072 and the initial velocity of the stone, is 13 meters per second.

03:02 So the magnitude of the initial momentum of the stone would be 0 .936 kilogram meters per second.

03:13 So that is this vector here...

Please give Ace some feedback