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 below). The 0.094-kg stone strikes the boat at a velocity of 14 m/s, 15° below due east, and ricochets off at a velocity of 10 m/s, 12° above due east. After being struck by the stone, the boat's velocity is 2.2 m/s, due east. What is the mass of the boat? Assume the water offers no resistance to the boat's motion.
Answer will be in kg.
Change of the horizontal component of the stone’s linear momentum is
Δp(x) = m•v(0x) – (-mv(1x)} = m{v(ox)+v(1x)} = m{v(0) •cos15 +v1•cos12}
For the boat Δp=p2-p1 = M•u – 0 = M•u.
Thje law of conservation of linear momentum
Δp(x)= Δp
m{v(0) •cos15 +v1•cos12}= M•u.
M = m{v(0) •cos15 +v1•cos12}/u=
=0.094(14•cos15+10•cos12)/2.2 = 1 kg
Ms.Vfx(stone)+Mb.Vf(boat)= Ms.Vox(stone)+Mb.Vo(boat)
Vo(boat)=0 m/s
So, we have:
Ms.Vfx(stone)+Mb.Vf(boat)= Ms.Vox(stone)
Mb.Vf(boat)= Ms.Vox(stone)-Ms.Vfx(stone)
Mb.Vf(boat)=Ms[Vox(stone)-Vfx(stone)]
Mb=Ms[Vox(stone)-Vfx(stone)/Vf(boat)
To solve this problem, we can use the principle of conservation of momentum. Conservation of momentum states that the total momentum before an event is equal to the total momentum after the event, as long as there are no external forces acting on the system.
Let's break down the problem step-by-step:
1. Convert all given velocities into their horizontal and vertical components.
- The initial velocity of the stone:
Initial horizontal velocity = 14 m/s * cos(15°) = 13.565 m/s
Initial vertical velocity = 14 m/s * sin(15°) = 3.731 m/s
- The final velocity of the stone after ricocheting:
Final horizontal velocity = 10 m/s * cos(12°) = 9.815 m/s
Final vertical velocity = 10 m/s * sin(12°) = 2.079 m/s
- The final velocity of the boat:
Final horizontal velocity = 2.2 m/s
2. Set up the conservation of momentum equation in the horizontal direction.
Before the collision:
(mass of the stone * initial horizontal velocity of the stone) + (mass of the boat * 0) = (mass of the stone * final horizontal velocity of the stone) + (mass of the boat * final horizontal velocity of the boat)
(0.094 kg * 13.565 m/s) + (mass of the boat * 0) = (0.094 kg * 9.815 m/s) + (mass of the boat * 2.2 m/s)
3. Simplify and solve for the mass of the boat.
1.276 m/s + 0 = 0.924 m/s + 2.2 m/s * mass of the boat
2.2 m/s * mass of the boat = 0.924 m/s - 1.276 m/s
2.2 m/s * mass of the boat = -0.352 m/s
mass of the boat = -0.352 m/s / 2.2 m/s
mass of the boat = -0.16 kg
Since a negative mass does not make physical sense, we can conclude that there is an error in either the given data or the calculations. Please double-check the given information and try again.
To find the mass of the boat, we can use the principle of conservation of momentum. According to this principle, the total momentum before the collision is equal to the total momentum after the collision.
The momentum of an object can be calculated by multiplying its mass by its velocity. In this case, we have the momentum of the stone before the collision and the momentum of the stone and the boat after the collision.
The momentum of the stone before the collision can be calculated as follows:
Momentum of the stone before = (Mass of the stone) * (Velocity of the stone before)
In this case, we have the mass of the stone (0.094 kg) and the velocity of the stone before the collision (14 m/s, 15° below due east).
Next, we need to calculate the momentum of both the stone and the boat after the collision. To do this, we will first break down the velocities into their horizontal and vertical components.
The horizontal component of the velocity is given by v * cos(θ), where v is the magnitude of the velocity and θ is the angle with respect to the horizontal axis.
The vertical component of the velocity is given by v * sin(θ), where v is the magnitude of the velocity and θ is the angle with respect to the horizontal axis.
For the stone after the collision, the horizontal and vertical components of the velocity are:
Horizontal component = 10 m/s * cos(12°)
Vertical component = 10 m/s * sin(12°)
For the boat after the collision, the horizontal and vertical components of the velocity are:
Horizontal component = 2.2 m/s * cos(0°) (since it is moving due east)
Vertical component = 2.2 m/s * sin(0°) (since it is moving due east)
Next, we calculate the momentum of the stone and the boat after the collision using the respective masses and velocities.
Momentum of the stone after = (Mass of the stone) * (Velocity of the stone after)
Momentum of the boat after = (Mass of the boat) * (Velocity of the boat after)
Now, we can set up an equation using the principle of conservation of momentum:
(Momentum of the stone before) = (Momentum of the stone after) + (Momentum of the boat after)
Finally, we can solve this equation for the mass of the boat.
Let's plug in the given values:
Momentum of the stone before = (0.094 kg) * (14 m/s, 15° below due east)
Momentum of the stone after = (0.094 kg) * (10 m/s, 12° above due east)
Momentum of the boat after = (Mass of the boat) * (2.2 m/s, due east)
By rearranging and solving the equation, we can find the mass of the boat.