A body is projected with a velocity of 200m/s at an angle of 30 degrees above the horizontal. calculate the time taken to reach the maximum height, its velocity after 16s

haw do you going calculate the time reach the maximum

To calculate the time taken to reach the maximum height, we can use the following formula:

time taken to reach maximum height = initial vertical velocity / acceleration due to gravity

Since the body is projected at an angle of 30 degrees above the horizontal, we first need to find the initial vertical velocity.

The initial vertical velocity can be calculated by multiplying the initial velocity by the sine of the angle.

Vertical velocity (Vy) = initial velocity (V) * sin(angle)

Vy = 200 * sin(30)

Vy = 100 m/s

Now, we can calculate the time taken to reach the maximum height.

time taken to reach maximum height = Vy / acceleration due to gravity

Using the standard value for the acceleration due to gravity, which is approximately 9.8 m/s²:

time taken to reach maximum height = 100 / 9.8

time taken to reach maximum height ≈ 10.2 seconds

Now, to calculate the velocity after 16 seconds, we can use the following formula:

velocity after certain time (Vt) = initial velocity (V) + acceleration (a) * time (t)

The acceleration in the vertical direction is equal to the acceleration due to gravity, which is -9.8 m/s² (negative sign indicates downward direction).

Vt = V + a * t

Vt = 200 + (-9.8) * 16

Vt = 200 - 156.8

Vt ≈ 43.2 m/s

Therefore, the velocity after 16 seconds is approximately 43.2 m/s.

To calculate the time taken to reach the maximum height, we can use the formula for the vertical component of projectile motion.

The initial velocity of the body in the vertical direction can be found by multiplying the initial velocity (200 m/s) by the sine of the launch angle (30 degrees).
Vertical initial velocity (Vy) = 200 m/s * sin(30 degrees)

To find the time taken to reach the maximum height, we need to calculate the time it takes for the vertical velocity to become zero. At the maximum height, the vertical velocity will be zero because the body is momentarily at rest and starts falling back down.

We can use the equation for vertical displacement to find this time, since the final vertical displacement will be zero at the maximum height.

The equation for vertical displacement is:
Vertical displacement (Sy) = Vy * t - 0.5 * g * t^2

Where g is the acceleration due to gravity, which is approximately 9.8 m/s^2.

Setting Sy = 0 and solving for t, we can find the time taken (t) to reach the maximum height.

Now, let's calculate:

Vertical initial velocity (Vy) = 200 m/s * sin(30 degrees)
Vy = 100 m/s

Using the equation for vertical displacement, setting Sy = 0:

0 = 100 m/s * t - 0.5 * 9.8 m/s^2 * t^2

Simplifying the equation:

0 = 100t - 4.9t^2

Rearranging the equation:

4.9t^2 - 100t = 0

Factoring out t:

t(4.9t - 100) = 0

Solving for t, we have two possible solutions:

t = 0 (initial time)
4.9t - 100 = 0

Solving for t, we find:

4.9t = 100
t = 100 / 4.9
t ≈ 20.4 seconds

So, it takes approximately 20.4 seconds for the body to reach the maximum height.

To calculate the velocity of the body after 16 seconds, we can determine the horizontal and vertical components of the velocity.

The horizontal component of the velocity (Vx) remains constant throughout the projectile motion. It can be found by multiplying the initial velocity (200 m/s) by the cosine of the launch angle (30 degrees).
Horizontal velocity (Vx) = 200 m/s * cos(30 degrees)

The vertical component of the velocity (Vy) changes over time due to gravitational acceleration. We can find the final vertical velocity after 16 seconds by subtracting the product of the acceleration due to gravity (9.8 m/s^2) and the time from the initial vertical velocity (Vy).

Final vertical velocity (Vy') = Vy - g * t

Now, let's calculate:

Horizontal velocity (Vx) = 200 m/s * cos(30 degrees)
Vx ≈ 173.2 m/s

Final vertical velocity (Vy') = 100 m/s - 9.8 m/s^2 * 16 s
Vy' ≈ -34.8 m/s

The negative sign indicates that the velocity is directed downwards.

Therefore, the velocity of the body after 16 seconds is approximately 173.2 m/s horizontally and -34.8 m/s vertically.