what speed must you toss a ball straight up so it takes 2s to return to you
To determine the speed at which you must toss a ball straight up so that it takes 2 seconds to return to you, we can use the equations of motion and the principles of projectile motion.
When an object is thrown straight up in the air and then falls back down under the influence of gravity, the motion can be divided into two parts: the upward motion and the downward motion. In the case of this question, the time taken for both parts of the motion is equal, which is 2 seconds.
First, let's consider the upward motion of the ball. The initial velocity (u) will be positive because it is in the upward direction. The final velocity (v) will be zero at the highest point because the ball momentarily comes to rest before falling back down. The acceleration due to gravity (g) will act in the downward direction, so it will be negative.
Using the equation of motion:
v = u + at
Since the final velocity in the upward motion is zero, we have:
0 = u - gt
Rearranging the equation to solve for u, we get:
u = gt
Now, let's consider the downward motion of the ball. The initial velocity (u') will be zero because the ball momentarily comes to rest at the highest point. The final velocity (v') will be negative because it is in the downward direction. The acceleration due to gravity (g) will act in the downward direction, so it will be negative.
Using the equation of motion:
v' = u' + at
Since the initial velocity in the downward motion is zero, we have:
v' = -gt
Using the equation of motion to find the position of the ball at any given time:
s = ut + (1/2)at^2
The total time for the upward and downward motions is 2 seconds, so we can set t = 2 seconds in the equation and substitute the values for u, v, u', and v':
0 = u - g(2)
0 = -gt - g(2)
Simplifying the equations, we find:
u = 2g
g = 9.8 m/s^2 (acceleration due to gravity)
Substituting the value of g:
u = 2(9.8)
u = 19.6 m/s
Therefore, you would need to toss the ball straight up with a speed of 19.6 m/s for it to take 2 seconds to return to you.