In a stunt for an action movie, the 100 kg actor jumps from a train which is crossing a river bridge. On the river below, the heroine is tied to a raft floating towards a waterfall at 3 m s−1. The raft and heroine have a total mass of 200 kg.
a If the hero times his jumps perfectly so as to land on the raft, and his velocity is 12 m s−1 at an angle of 80° to the river current, what will be the velocity of the raft immediately after he lands? Draw a vector diagram to show the momentum addition. (Ignore any vertical motion.)
b If the waterfall is 100 m downstream, and the hero landed when the raft was 16 m from the bank, would they plummet over the fall? (Assume the velocity remains constant after the hero has landed.)
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a) To find the velocity of the raft immediately after the hero lands, we need to consider the momentum addition from the hero and the current.
Let's start by finding the horizontal component of the hero's velocity. We can use the given velocity and angle:
Horizontal component of hero's velocity = 12 m/s * cos(80°)
= 12 m/s * 0.1736
= 2.08 m/s
Now let's find the horizontal component of the river current. We are given the speed of the current, but we need to find its horizontal component:
Horizontal component of river current = 3 m/s * cos(0°)
= 3 m/s * 1
= 3 m/s
Next, we find the total horizontal momentum after the hero lands. The mass of the hero is 100 kg, and the mass of the raft with the heroine is 200 kg:
Total mass = mass of hero + mass of raft = 100 kg + 200 kg = 300 kg
Total horizontal momentum = (mass of hero * horizontal component of hero's velocity) + (mass of raft * horizontal component of river current)
= (100 kg * 2.08 m/s) + (200 kg * 3 m/s)
= 208 kg·m/s + 600 kg·m/s
= 808 kg·m/s
Finally, we can find the velocity of the raft immediately after the hero lands by dividing the total horizontal momentum by the total mass:
Velocity of the raft = Total horizontal momentum / Total mass
= 808 kg·m/s / 300 kg
= 2.69 m/s
Therefore, the velocity of the raft immediately after the hero lands will be 2.69 m/s (approximately).
b) To determine if they will plummet over the waterfall, we need to compare the horizontal distance between the landing point and the bank (16 m) with the downstream distance to the waterfall (100 m).
If the hero lands on the raft when it is 16 m from the bank, and the raft and heroine continue moving downstream at a constant velocity after the hero lands, the raft will travel another 100 m to reach the waterfall.
Since the horizontal distance to the waterfall (100 m) is greater than the distance from the landing point to the bank (16 m), the raft and the heroine will indeed plummet over the waterfall.
To find the velocity of the raft immediately after the hero lands, we need to consider the momentum before and after his jump.
a) The momentum before the hero jumps can be calculated by multiplying his mass (100 kg) by his velocity (12 m/s) in the x-direction:
Momentum before = mass × velocity = 100 kg × 12 m/s = 1200 kg·m/s
Since the hero's velocity is at an angle of 80° to the river current, we can break it down into its x and y components using trigonometry. The x-component of the hero's velocity can be found by multiplying the total velocity (12 m/s) by the cosine of the angle (80°):
Velocity (x-component) = 12 m/s × cos(80°) = 2.347 m/s
Since the hero is jumping onto a stationary raft, the initial momentum of the raft and the heroine is zero.
The momentum after the hero lands on the raft can be calculated by multiplying the total mass of the raft and heroine (200 kg) by their common velocity:
Momentum after = mass × velocity = 200 kg × velocity (after landing)
Since momentum is conserved, the total momentum before and after the hero's jump must be equal. Therefore:
Momentum before = Momentum after
1200 kg·m/s = 200 kg × velocity (after landing)
Solving for velocity (after landing), we get:
velocity (after landing) = 1200 kg·m/s / 200 kg = 6 m/s
So, the velocity of the raft immediately after the hero lands is 6 m/s.
b) To determine if the raft and heroine will plummet over the fall, we need to consider their displacement and the time it takes for the raft to reach the waterfall.
Given that the waterfall is 100 m downstream and the raft is initially 16 m from the bank, the total displacement of the raft is:
Displacement = 100 m + 16 m = 116 m
Since the velocity of the raft is 6 m/s, we can use the formula:
velocity = displacement / time
To find the time it takes for the raft to reach the waterfall:
time = displacement / velocity = 116 m / 6 m/s = 19.33 s
If the velocity remains constant after the hero has landed, and it takes 19.33 seconds for the raft to reach the waterfall, then the raft and heroine will indeed plummet over the fall since they will not reach the bank in time.