a stone that is thrown near the surface of the earth. There is no air resistance. Position increases in the up direction. The initial velocity is 30 m/s at 30 degrees above the horizontal.

what is the velocity when the stone reaches its maximum height ?
what is the acceleration when it reaches its maximum height
what is the time required for the stone to reach the maximum height ?

Vi = 30 sin 30 = 15 = initial speed UP

u = 30 cos 30 = 26 = horizontal speed until caught

v = Vi - 9.81 t
at top v = 0 (First answer)
so at top
t = 15/9.81 = 1.53 seconds upward bound

THE ACCELERATION IS -9.81 m/s^2 the ENTIRE TIME that it is airborne

To answer these questions, we need to break down the motion of the stone and use the laws of physics.

First, let's find the velocity when the stone reaches its maximum height:

1. The initial velocity of the stone can be divided into its horizontal and vertical components. The horizontal component is given by: Vx = V * cos(theta), where V is the initial velocity (30 m/s) and theta is the angle above the horizontal (30 degrees).

Vx = 30 m/s * cos(30 degrees) = 25.98 m/s (rounded to two decimal places)

2. Since there is no air resistance, the vertical component of the stone's velocity will change due to gravity. The vertical component of the velocity at maximum height will be zero. We can use this information to find the time it takes for the stone to reach its maximum height.

The equation for calculating time is given by: t = Vy / g, where Vy is the vertical component of the initial velocity and g is the acceleration due to gravity on Earth (approximately 9.8 m/s^2).

Vy = V * sin(theta) = 30 m/s * sin(30 degrees) = 15 m/s (rounded to two decimal places)

t = 15 m/s / 9.8 m/s^2 ≈ 1.53 seconds (rounded to two decimal places)

3. Now that we know the time it takes for the stone to reach its maximum height, we can find the vertical component of the stone's velocity at this point. Velocity is given by: V = V₀ + a * t, where V is the final velocity, V₀ is the initial velocity, a is the acceleration, and t is the time.

At maximum height, the vertical component of the velocity is zero, so the equation can be simplified as follows:

0 = 15 m/s + (-9.8 m/s^2) * 1.53 seconds

Solving for V gives us: V ≈ -15 m/s

The negative sign indicates that the velocity is pointing in the opposite direction of the initial vertical velocity (upward).

Therefore, the velocity when the stone reaches its maximum height is approximately -15 m/s (upward).

Next, let's find the acceleration when the stone reaches its maximum height:

Since the stone is at its highest point, its vertical velocity is zero. At this point, the only force acting on the stone is gravity, which pulls it downward. Therefore, the acceleration at maximum height is equal to the acceleration due to gravity on Earth, which is approximately 9.8 m/s^2.

Thus, the acceleration when the stone reaches its maximum height is approximately 9.8 m/s^2.

Finally, let's calculate the time required for the stone to reach the maximum height:

As mentioned earlier, the time it takes for the stone to reach its maximum height is approximately 1.53 seconds.

Therefore, the time required for the stone to reach the maximum height is approximately 1.53 seconds.