a ball is projected horizontally with speed u from a tower. 4 seconds after the throw it's velocity is at 45¡ã with horizontal. How long after projection, the velocity is at 37¡ã with horizontal?
To solve this problem, we will use the principles of projectile motion.
First, let's analyze the situation. A ball is projected horizontally from a tower, which means its initial velocity in the vertical direction is zero. The ball experiences constant acceleration in the vertical direction due to gravity, but because it was launched horizontally, there is no horizontal acceleration.
Now, let's find the time at which the velocity is at 45° with the horizontal. We know that when the velocity is at 45° with the horizontal, the vertical and horizontal components of the velocity are equal. Since the initial vertical velocity is zero, this means that the vertical velocity 4 seconds after the throw is equal to the horizontal velocity at that time.
To find the vertical velocity 4 seconds after the throw, we can use the kinematic equation:
vf = vi + at
Since the initial vertical velocity (vi) is zero, and the acceleration (due to gravity) is -9.8 m/s^2 (assuming the upward direction as positive), the equation becomes:
vf = 0 + (-9.8 * 4)
Simplifying, we find that the vertical velocity 4 seconds after the throw is -39.2 m/s (assuming downward as negative).
Since the vertical and horizontal velocities are equal, the horizontal velocity at that time is also -39.2 m/s.
Now, let's find the time at which the velocity is at 37° with the horizontal. We can use the same approach as before. We know that the vertical velocity at that time is equal to the horizontal velocity at that time:
vf = 0 + (-9.8 * t)
Furthermore, we know that the tangent of the angle (37°) is equal to the vertical velocity divided by the horizontal velocity:
tan(37°) = vf / (-39.2)
Rearranging the equation, we find:
vf = -39.2 * tan(37°)
Now, we can equate the two expressions for vf:
-39.2 * 4 = -39.2 * tan(37°) * t
Simplifying, we find:
4 = tan(37°) * t
Finally, solving for t:
t = 4 / tan(37°)
Using a calculator, we find that t is approximately 3.29 seconds.
Therefore, the velocity is at 37° with the horizontal approximately 3.29 seconds after the projection.