1. A chunk of rock of mass m breaks loose from the top of a rock climber’s dome and falls for time t.

Question

a. Neglecting air resistance, show that its momentum when it hits the ground below is mgt.

b. If it falls from a higher dome where the falling time is 2 t, show that its momentum when hitting the ground would be 2 mgt.

c. If the falling rock were more massive, how would this affect falling time?

d. Show that the height of a dome corresponding to a 3-second drop is 45 m. (Use the acceleration of free fall, [latex]g = 10 m/s^2[/latex] .)

2. An ostrich egg of mass m is thrown at speed v into a sagging bed sheet. The egg is brought to rest in time t.

a. Show that the average force of egg impact is [latex]\frac{mv}{t} [/latex] .

b. If the mass of the egg is 1.0 kg, its speed when hitting the sheet is 2.0 m/s, and it is brought to rest in 0.2 s, show that the average force of impact is 10 N.

c. Why is breakage less likely with a sagging sheet than with a taut one?

Accepted Answer

1. a. Its momentum hitting the ground is its [latex]mass imes speed[/latex] . The falling speed from rest is v = gt. So the momentum after time t is mv = mgt.

b. Momentum = ?

Twice the time in the air [latex]\Longrightarrow[/latex] twice the impulse on the chunk of rock [latex]\Longrightarrow[/latex] twice the change in momentum. So the final momentum would be twice, 2 mgt.

c. Except for the effects of air resistance, falling time doesn’t depend on mass. The rock would have greater momentum due to its greater mass, but not due to greater speed.

d. When falling from rest, [latex]d = 1/2 gt^2= 1/2 (10 m/s^2)(3 s)^2 = 45 m.[/latex] .

2. a. From the impulse–momentum equation, [latex]F t=\Delta m v[/latex], where in this case the egg ends up at rest, [latex]\Delta m v=m v[/latex], and simple algebraic rearrangement gives [latex]F = \frac{mv}{t}[/latex] .

b. [latex]F=\frac{m v}{t}=\frac{(1.0 kg )\left(2.0 \frac{ m }{ s } ight)}{(0.2 s )}=10 kg \cdot \frac{ m }{ s ^{2}}=10 N[/latex].

c. The time during which the tossed egg’s momentum goes to zero is extended when it hits a sagging sheet. Extended time means less force in the impulse that brings the egg to a halt. Less force means less chance of breakage.

Question 5.PS.1: 1. A chunk of rock of mass m breaks loose from the top of a ...

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