A ball is dropped from a cliff and 2 seconds later, a second ball is thrown vertically downward. What must be the initial velocity of the second ball if it overtakes the first ball as it passes 400m ?
If you mean that the balls pass as they have both fallen 400m, then you have
4.9t^2 = 400
v(t-2)+4.9(t-2)^2 = 400
v(20/√4.9-2)+4.9(20/√4.9-2)^2 = 400
v = 22.39 m/s
To determine the initial velocity of the second ball, we need to consider the motion of both balls. Let's break down the problem into steps:
Step 1: Starting with the first ball, we need to determine its position as a function of time using the equation of motion.
The equation for the position of an object in freefall is given by:
s = ut + (1/2)gt²
where s is the position, u is the initial velocity, t is the time, and g is the acceleration due to gravity (-9.8 m/s²).
Given that the first ball is dropped from rest (initial velocity, u1 = 0), the equation simplifies to:
s1 = (1/2)gt²
Step 2: Solve for the position of the first ball when the second ball is thrown.
The second ball is thrown 2 seconds after the first ball is dropped. Therefore, we need to find the position of the first ball after 2 seconds.
Substituting t = 2 seconds into the equation:
s1 = (1/2)(-9.8 m/s²)(2 s)²
s1 = -19.6 m/s² * 4 s²
s1 = -78.4 m
So, after 2 seconds, the position of the first ball is 78.4 meters below the starting point.
Step 3: Determine the equation for the second ball.
The equation for the position of the second ball can be written as:
s2 = u2t + (1/2)gt²
where u2 is the initial velocity of the second ball.
Step 4: Find the initial velocity of the second ball.
To find the initial velocity of the second ball when it overtakes the first ball, we need to set the positions of the two balls equal to each other and solve for u2:
s2 = s1
u2t + (1/2)gt² = -78.4 m
u2(2 s) + (1/2)(-9.8 m/s²)(2 s)² = -78.4 m
2u2 - 19.6 m/s² * 4 s² = -78.4 m
2u2 - 78.4 m/s² = -78.4 m
2u2 = 0
u2 = 0 m/s
From the calculations, we find that the initial velocity of the second ball needs to be 0 m/s for it to overtake the first ball as it passes 400m.