After skiding down a snow-covered hill on an inner tube, Ashley is coasting across a level snowfield at a constant velocity of +2.7 m/s. Miranda runs after her at a velocity of +5.0 m/s and hops on the inner tube. How fast do the two of them slide across the snow together on the inner tube? Ashley's mass is 61 kg, and Miranda's is 59 kg. Ignore the mass of the inner tube and any friction between the inner tube and the snow.
what formula do i use
To determine the speed at which Ashley and Miranda slide across the snow together on the inner tube, we can apply the principle of conservation of momentum. The formula to use is:
Total momentum before = Total momentum after
In this case, the total momentum before Miranda hops on the inner tube is equal to the momentum of Ashley alone. After she joins Ashley on the inner tube, the total momentum is the sum of their individual momenta.
The formula can be written as:
(mass of Ashley × velocity of Ashley) + (mass of Miranda × velocity of Miranda) = (total mass × final velocity)
Let's substitute the given values into the equation:
(61 kg × 2.7 m/s) + (59 kg × 5.0 m/s) = (61 kg + 59 kg) × final velocity
Simplifying further:
164.7 kg⋅m/s + 295 kg⋅m/s = 120 kg × final velocity
459.7 kg⋅m/s = 120 kg × final velocity
Dividing both sides by 120 kg gives:
final velocity = 459.7 kg⋅m/s / 120 kg
final velocity ≈ 3.83 m/s
Therefore, when Miranda joins Ashley on the inner tube, they will slide across the snow together at a speed of approximately 3.83 m/s.
To solve this problem, we can use the principle of conservation of momentum. According to this principle, the total momentum before the two individuals combine on the inner tube should be equal to the total momentum after they combine.
The formula for momentum is given by:
Momentum = mass x velocity
In this case, Ashley and Miranda combine on the inner tube, so the total mass is the sum of their individual masses:
Total mass = Ashley's mass + Miranda's mass
The total momentum before they combine is given by:
Initial momentum = Ashley's momentum + Miranda's momentum
Ashley's momentum = Ashley's mass x Ashley's velocity
Miranda's momentum = Miranda's mass x Miranda's velocity
The total momentum after they combine is given by:
Final momentum = Total mass x Final velocity
Since the final velocity is the unknown in this case, we can rearrange the equation to solve for it:
Final velocity = Final momentum / Total mass
In summary:
1. Calculate Ashley's momentum using Ashley's mass and velocity.
2. Calculate Miranda's momentum using Miranda's mass and velocity.
3. Add Ashley's and Miranda's momentum to find the initial momentum.
4. Calculate the total mass using Ashley's and Miranda's mass.
5. Divide the initial momentum by the total mass to find the final velocity.