An inquisitive physics student and mountain climber climbs a 53.0-m-high cliff t
ID: 1636910 • Letter: A
Question
An inquisitive physics student and mountain climber climbs a 53.0-m-high cliff that overhangs a calm pool of water. He throws two stones vertically downward, 1.00 s apart, and observes that they cause a single splash. The first stone has an initial speed of 1.96 m/s. (a) How long after release of the first stone do the two stones hit the water? ________ s (b) What initial velocity must the second stone have if the two stones are to hit the water simultaneously? magnitude _________ m/s direction (c) What is the speed of each stone at the instant the two stones hit the water? first stone _________ m/s second stone ___________ m/sExplanation / Answer
(a) Here, the two stones is thrown vertically downward, 1.00 s apart, and observes that they cause a single splash.
Now, consider the first stone takes time t, the second stone takes time t-1.
Again, the first stone has an initial speed of 1.96 m/s.
So the equation governing its motion -
d = v0*t + (1/2)*at^2 -----------------------------------------(i)
given that d = 53 m and v0 = 1.96 m/sec, and a = acceleration due to gravity, 9.8 m/sec^2.
On the other hand, the second stone follows this equation:
d = v1*(t-1) + (1/2)*a(t-1)^2 ---------------------------------------------(ii)
since the time of travel is t-1 and the initial velocity is v1 (unknown). The distance d is the same 53 m.
Well, t is the only unknown for the first stone. So just solve
d = v0*t + (1/2)*at^2
=> 53 = 1.96t + (1/2)*9.8 t^2
=> 4.9t^2 + 1.96t - 53 = 0
=> t = [-1.96 + sqrt(1.96^2 + 4*4.9*53)] / (2*4.9) = [-1.96 + 32.29] / 9.8 = 3.09 s.
The other value of t is negative so you can discard that value.
So, your answer is t = 3.09 s.
(b) put the value of t in the second equation -
53 = v1 (3.09 - 1) + 0.5*9.8*(3.09-1)^2
=> 53 - 21.40 = 2.09 * v1
=> v1 = 15.12 m/s vertically upward.
(c) for the first stone -
final velocity, v = u + a*t = 1.96 - 9.8*3.09 = -28.32 m/s.
The negative sign shows that its direction is in downward.
For the second stone -
v = 15.12 - 9.8*2.09 = - 5.36 m/s.
Its direction is also in downward.