A stone thrown horizontally from the top of a 18-m tower hits the ground at a point 18 m from the base of the tower. (Ignore any effects due to air resistance.)
(a) Find the speed at which the stone was thrown.
m/s
(b) Find the speed of the stone just before it hits the ground.
m/s
See previous post: Wed, 12-3-14, 8:20 AM
To solve this problem, we can apply the principles of projectile motion and the equations of motion.
(a) Find the speed at which the stone was thrown.
To find the initial speed at which the stone was thrown, we can use the equation for horizontal motion:
distance = speed × time
In this case, the distance traveled horizontally by the stone is 18 meters, and we know that the distance is equal to the time multiplied by the horizontal component of the initial velocity.
Since the stone is thrown horizontally, the initial vertical velocity is 0 m/s, and the initial horizontal velocity is what we are interested in finding.
Using the equation for horizontal motion, we have:
18 meters = speed × time
Since the initial vertical velocity is 0 m/s, the time it takes for the stone to hit the ground can be found using the equation for vertical motion:
distance = (initial velocity × time) + (0.5 × acceleration × time^2)
In this case, the distance traveled vertically is the height of the tower, which is 18 meters, the initial vertical velocity is 0 m/s, and the acceleration due to gravity is -9.8 m/s^2 (negative because it acts downward).
Substituting the known values, we have:
18 meters = 0 × time + (0.5 × -9.8 m/s^2 × time^2)
Simplifying:
18 meters = -4.9 m/s^2 × time^2
Rearranging the equation:
time^2 = (18 meters) / (-4.9 m/s^2)
time^2 ≈ -3.67 seconds^2 (ignoring the negative solution since time cannot be negative)
Taking the square root:
time ≈ 1.92 seconds
Substituting the time value back into the equation for horizontal motion:
18 meters = speed × 1.92 seconds
Simplifying:
speed ≈ 18 meters / 1.92 seconds
speed ≈ 9.38 m/s
Therefore, the speed at which the stone was thrown is approximately 9.38 m/s.
(b) Find the speed of the stone just before it hits the ground.
To find the speed of the stone just before it hits the ground, we can use the equation for vertical motion:
final velocity = initial velocity + (acceleration × time)
In this case, the initial vertical velocity is 0 m/s, the acceleration due to gravity is -9.8 m/s^2, and the time we obtained earlier is approximately 1.92 seconds.
Substituting the known values, we have:
final velocity = 0 m/s + (-9.8 m/s^2 × 1.92 seconds)
Simplifying:
final velocity ≈ -18.816 m/s
Since the stone is moving in the downward direction, the final velocity is negative. Taking the absolute value, the speed of the stone just before it hits the ground is approximately 18.816 m/s.
Therefore, the speed of the stone just before it hits the ground is approximately 18.816 m/s.