After sliding down a snow-covered hill on an inner tube, Ashley is coasting across a level snowfield at a constant velocity of +2.8 m/s. Miranda runs after her at a velocity of +4.8 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 69 kg and Miranda's is 61 kg. Ignore the mass of the inner tube and any friction between the inner tube and the snow.
Well, let's do some math while trying not to slip on the snowy slope of physics!
We know that Ashley is already coasting at a constant velocity of +2.8 m/s. Now, Miranda hops on the inner tube with her speedy legs and adds her velocity to the mix. Miranda has a velocity of +4.8 m/s.
To find the total velocity of both of them together, we add their individual velocities: 2.8 m/s + 4.8 m/s = 7.6 m/s.
So, the two of them slide across the snow together on the inner tube at a dazzling speed of 7.6 m/s. That's like a snowflake on steroids!
To solve this problem, we can use the concept of conservation of momentum. According to this principle, the total momentum before the two individuals hop on the inner tube should be equal to the total momentum after they are on the inner tube.
The momentum of an object is calculated by multiplying its mass by its velocity. Therefore, the momentum before Ashley and Miranda jump on the inner tube is given by:
momentum_before = (Ashley's mass × Ashley's velocity) + (Miranda's mass × Miranda's velocity)
momentum_before = (69 kg × 2.8 m/s) + (61 kg × 4.8 m/s)
Now, let's calculate this value:
momentum_before = 193.2 kg·m/s + 292.8 kg·m/s
= 486 kg·m/s
The total momentum after they are on the inner tube should also be equal to 486 kg·m/s.
Since Ashley and Miranda are traveling together on the inner tube, their combined mass is equal to the sum of their individual masses:
total_mass = Ashley's mass + Miranda's mass
total_mass = 69 kg + 61 kg
= 130 kg
The final step is to calculate the velocity at which they slide across the snow together. This can be found using the formula:
velocity_together = total_momentum / total_mass
velocity_together = 486 kg·m/s / 130 kg
Now, let's calculate this value:
velocity_together = 3.74 m/s
Therefore, Ashley and Miranda will slide across the snow together on the inner tube at a constant velocity of 3.74 m/s.
To determine the speed at which Ashley and Miranda slide across the snow together on the inner tube, we can use the principle of conservation of momentum.
The momentum of an object is the product of its mass and velocity. The momentum before Miranda hops on the inner tube is equal to the momentum after she hops on.
Let's first calculate the momentum of Ashley before Miranda jumps on:
Momentum of Ashley = Mass of Ashley × Velocity of Ashley.
Momentum of Ashley = 69 kg × (+2.8 m/s) = 193.2 kg·m/s (since the velocity is in the positive direction).
Now, let's calculate the momentum of Miranda before she jumps onto the inner tube:
Momentum of Miranda = Mass of Miranda × Velocity of Miranda.
Momentum of Miranda = 61 kg × (+4.8 m/s) = 292.8 kg·m/s.
According to the conservation of momentum, the total momentum before and after Miranda jumps onto the inner tube should be the same.
Total momentum before = Total momentum after.
(69 kg × Velocity of Ashley) + (61 kg × Velocity of Miranda) = (69 kg + 61 kg) × Final Velocity.
Let's solve the equation to find the Final Velocity:
(69 kg × 2.8 m/s) + (61 kg × 4.8 m/s) = (69 kg + 61 kg) × Final Velocity.
193.2 kg·m/s + 292.8 kg·m/s = 130 kg × Final Velocity.
485.9 kg·m/s = 130 kg × Final Velocity.
Divide both sides of the equation by 130 kg to solve for Final Velocity:
Final Velocity = 485.9 kg·m/s / 130 kg.
Final Velocity = 3.74 m/s.
Therefore, the speed at which Ashley and Miranda slide across the snow together on the inner tube is 3.74 m/s.