Question

A projectile is fired from ground level at an angle of 40.0° above horizontal at a speed of 30.0 m/s. What is the speed of the projectile when it has reached a height equal to 50.0% of its maximum height?

Answer 1

26.74 m/s

Step-by-step explanation:

The vertical and horizontal of the velocity of the projectile when it first fired from ground level at an angle of 40 degrees

`v_v = vsin\theta = 30sin40^o = 19.3 m/s`

`v_o = vcos\theta = 30cos40^o = 23 m/s`

As the ball travels, vertically speaking its kinetic energy is converted into potential energy. When it reaches its maximum height the kinetic energy would be 0. Also let potential energy be 0 at around, we have the following energy conservation equation:

`E_k = E_p`

`mv_v^2/2 = mgH`

Let g = 9.81 m/s2. We can divide both sides by m

`v_v^2/2 = gH`

`H = v_v^2/2g = 19.3^2/(2*9.81) = 18.95 m`

So half of the maximum height is h = H/2 = 9.48 m

Using the same energy equation at h = 9.48 m then we have

`mv_v^2/2 = mv_h^2/2 + mgh`

Again, divide both sides by m, and multiply by 2

`v_v^2 = v_h^2 + 2gh`

`v_h^2 = v_v^2 - 2gh = 19.3^2 - 2*9.81*9.48 = 186`

`v_h = √(186) = 13.64 m/s`

If we ignore air resistance, then nothing could affect the horizontal velocity, it would stay the same at h = 9.48 m, 23m/s

`V = √(v_h^2 + v_o^2) = √(13.64^2 + 23^2) = √(186.0496 + 529) = √(715.0496) = 26.74m/s`

Answer 2

Answer: 26.7 m/s

Step-by-step explanation:

The formulae for the maximum height of a projectile motion is given as

H = u²× (sinx)²/2g

Where x is the angle of inclination = 40° and g is the acceleration due to gravity = 9.8m/s².

u = initial velocity of the projectile motion = 30 m/s

By substituting the parameters, we have that

Hmax = 30² × (sin40°)²/2×9.8

Hmax = 900 × 0.4132/ 19.6

Hmax = 371.858/ 19.6 = 18.9m

But the height (H) of the projectile is 50% of the maximum height.

H = 0.5× Hmax

H = 0.5 × 18.9

H = 9.5m

Recall that a projectile motion is defined in the vertical axis by the equation given below.

v² = u² - 2gH

v² = 30² - 2 ×9.8 × 9.5

v² = 900 - 185.93

v² = 714.07

v = √714.07

v = 26.7 m/s