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Chapter 2: Problem 18

You and a friend are standing at the edge of a snowcovered cliff. At the same time, you throw a snowball straight upward with a speed of \(8.0 \mathrm{~m} / \mathrm{s}\) over the edge of the cliff and your friend throws a snowball straight downward over the edge of the cliff with the same speed. Your snowball is twice as heavy as your friend's. Neglecting air resistance, which snowball will hit the ground first, and which will have the greater speed?

Short Answer

Expert verified
Answer: Both snowballs will hit the ground at the same time. Their velocities just before impact will be equal in magnitude but opposite in direction, as \(v_{Af} = -v_{Bf}\).

Step by step solution

01

Define the given values

Initial upward velocity of snowball A (thrown upward), \(v_{A0} = 8.0\) m/s Initial downward velocity of snowball B (thrown downward), \(v_{B0} = -8.0\) m/s Acceleration due to gravity, \(g = 9.8\) m/s²
02

Determine the time for each snowball to reach the ground

Both snowballs will reach the ground when their respective vertical positions is equal to the height of the cliff, assuming the height of the cliff is h. To determine their time of flight, we'll use the equation \(y = v_0t + \frac{1}{2}gt^2\), where \(y\) is the vertical displacement, \(v_0\) is the initial vertical velocity, and \(t\) is the time. For both snowballs, the equation will be \(h = v_0t + \frac{1}{2}gt^2\). Let's call the time for snowball A \(t_A\), and time for snowball B \(t_B\).
03

Solve those equations for each of the snowballs

For snowball A (thrown upward): \(h = 8.0t_A - 0.5(9.8)t_A^2\) For snowball B (thrown downward): \(h = (-8.0)t_B + 0.5(9.8)t_B^2\) Because they are thrown off the same cliff and are both subject to the same gravitational acceleration, \(t_A = t_B\). Thus, both snowballs will hit the ground at the same time.
04

Determine their velocities just before impact

To find their final velocities (just before impact), we'll use the equation \(v_f = v_0 + gt\). For snowball A (thrown upward): \(v_{Af} = 8.0 - 9.8t_A\) For snowball B (thrown downward): \(v_{Bf} = -8.0 + 9.8t_B\) Since \(t_A = t_B\), their final velocities just before impact will be equal in magnitude but opposite in direction: \(v_{Af} = -v_{Bf}\)
05

Conclusion

Both snowballs will hit the ground at the same time, with equal magnitude velocities but opposite directions. The mass of the snowballs does not affect the time of flight or their velocity upon impact, but it does affect their momentum. The heavier snowball (thrown upward) will have twice the momentum of the lighter snowball (thrown downward) when they hit the ground.

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Most popular questions from this chapter

The fastest speed in NASCAR racing history was \(212.809 \mathrm{mph}\) (reached by Bill Elliott in 1987 at Talladega). If the race car decelerated from that speed at a rate of \(8.0 \mathrm{~m} / \mathrm{s}^{2},\) how far would it travel before coming to a stop?

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