After skiding down a snow-covered hill on an inner tube, Ashley is coasting across a level snowfield at a constant velocity of +3.3 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 45 kg, and Miranda's is 59 kg. Ignore the mass of the inner tube and any friction between the inner tube and the snow
To find the combined velocity of Ashley and Miranda sliding on the inner tube, we can use the law of conservation of momentum.
The law states that the total momentum before an event is equal to the total momentum after the event, provided there are no external forces acting on the system.
The momentum p of an object is calculated by multiplying its mass m by its velocity v (p = mv).
Given that Ashley is already coasting across the snowfield at a constant velocity of 3.3 m/s, her momentum is calculated as follows:
Ashley's momentum = mass × velocity
= 45 kg × 3.3 m/s
= 148.5 kg·m/s (in the positive direction)
Similarly, Miranda's momentum while running after Ashley at 4.8 m/s is:
Miranda's momentum = mass × velocity
= 59 kg × 4.8 m/s
= 283.2 kg·m/s (in the positive direction)
Since there are no external forces acting on the system once Miranda hops on the inner tube, the total momentum before and after the event must be the same.
Total momentum before = Total momentum after
Therefore, the combined momentum of Ashley and Miranda can be calculated as:
Combined momentum = Ashley's momentum + Miranda's momentum
= 148.5 kg·m/s + 283.2 kg·m/s
= 431.7 kg·m/s (in the positive direction)
To find the combined velocity, we divide the combined momentum by the total mass of both individuals:
Combined velocity = Combined momentum / (Ashley's mass + Miranda's mass)
= 431.7 kg·m/s / (45 kg + 59 kg)
= 431.7 kg·m/s / 104 kg
= 4.152 m/s (in the positive direction)
Therefore, the combined velocity of Ashley and Miranda sliding on the inner tube is approximately 4.152 m/s in the positive direction.