After skiding down a snow-covered hill on an inner tube, Ashley is coasting across a level snowfield at a constant velocity of +3.4 m/s. Miranda runs after her at a velocity of +4.0 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 45 kg, and Miranda's is 46 kg. Ignore the mass of the inner tube and any friction between the inner tube and the snow.
momentum before = 45(3.4)+46(4)
momentum after = (91)v
so
91 v = 45(3.4)+46(4)
3.7 m/s
To find the speed at which Ashley and Miranda slide across the snow together on the inner tube, we need to consider the principle of conservation of momentum.
Conservation of momentum states that the total momentum of a system remains constant if no external forces act upon it. In this case, before Miranda hops on the inner tube, Ashley is already sliding across the snow at a constant velocity. This means that the momentum of Ashley and the inner tube must be conserved when Miranda jumps on.
Momentum, denoted by the symbol 'p,' is given by the product of an object's mass and velocity. Since we have the mass and velocity of Ashley and Miranda before they join, we can find their initial momentum individually and then add them together to find the total momentum.
The equation for momentum is:
momentum = mass × velocity
For Ashley alone:
momentum_Ashley = mass_Ashley × velocity_Ashley
For Miranda alone:
momentum_Miranda = mass_Miranda × velocity_Miranda
In this case, both Ashley and Miranda are moving in the positive direction, so their velocities can be taken as positive.
Next, we can find the total momentum after Miranda jumps on:
total_momentum = momentum_Ashley + momentum_Miranda
Now that we have the total momentum, we need to find the velocity at which Ashley and Miranda slide across the snow together. Since momentum is defined as mass times velocity, we can rearrange the equation to solve for velocity:
velocity = total_momentum / total_mass
In this case, the total mass is the sum of Ashley and Miranda's masses.
Finally, we can substitute the values that we have and calculate the final velocity.
Let's plug in the given values:
mass_Ashley = 45 kg
velocity_Ashley = +3.4 m/s
mass_Miranda = 46 kg
velocity_Miranda = +4.0 m/s
momentum_Ashley = 45 kg × 3.4 m/s = 153 kg·m/s
momentum_Miranda = 46 kg × 4.0 m/s = 184 kg·m/s
total_momentum = 153 kg·m/s + 184 kg·m/s = 337 kg·m/s
total_mass = 45 kg + 46 kg = 91 kg
velocity = 337 kg·m/s / 91 kg ≈ 3.7 m/s
Therefore, Ashley and Miranda will slide across the snow together on the inner tube at a velocity of approximately +3.7 m/s.