An object is shot at 20m/s at an angle theta. The object lands 24m away. What is the angle?

Can you check if I am doing this right?

I had
v(initial) = 20 m/s
g = 9.8 m/s^2
x = 24m

1st I solved for time:
x = v(initial) t
x = v(initial) cos(theta) t
24 = 20 cos(theta) t
24 / [20 cos(theta)] = t

Y-motion
y = v(initial) sin(theta)*t - (1/2)gt^2

0 = [20sin(theta)]*[24 / 20cos(theta)] t -(1/2)(9.8)[24 / 20cos(theta)]^2

0 = 24tan(theta) - (7.056/cos^2(theta))
7.056 = 24tan(theta)cos^2(theta)
7.056 = 24sin(theta)cos(theta)
<then I divided both sides by 12 to get a trig identity>
.588 = 2sin(theta)cos(theta)
.588 = sin2(theta)
theta = arcsin(.588) / 2
theta = 18.008 <--answer

Is that how you would do it? Is that the correct answer?

I will skip the derivation and cut to the final equation for horizontal range X:


X = (V^2/g)sin 2A
where A = the launch angle.

24 = (400/9.8) sin 2A
sin 2A = 0.588
2A = 36 or 144 degrees
A = 18 or 72 degrees

Looks like you did it right. Consider the possibility of two solutions.

Your approach to solving the problem is correct, but there seems to be a small mistake in your calculations.

Let's go over the steps again to find the correct answer.

Given:
Initial velocity (v(initial)) = 20 m/s
Gravitational acceleration (g) = 9.8 m/s^2
Horizontal distance (x) = 24 m

First, let's solve for time (t):
We know that the horizontal distance (x) is given by x = v(initial) * cos(theta) * t.
Plugging in the values, we have 24 = 20 * cos(theta) * t.

Dividing both sides of the equation by 20 * cos(theta), we get:
t = 24 / (20 * cos(theta)).

Now, let's substitute this value of t into the expression for the vertical motion.

The vertical position (y) at time t is given by y = v(initial) * sin(theta) * t - (1/2) * g * t^2.
Plugging in the values, we have:
y = 20 * sin(theta) * (24 / (20 * cos(theta))) - (1/2) * 9.8 * (24 / (20 * cos(theta)))^2.

Simplifying, we get:
y = 24 tan(theta) - 2.94 / cos^2(theta).

Now, let's rearrange this equation to solve for theta:
24 tan(theta) - 2.94 / cos^2(theta) = 0.

Multiplying through by cos^2(theta), we get:
24 sin(theta) - 2.94 = 0.

Rearranging further, we have:
24 sin(theta) = 2.94.

Dividing both sides by 24, we get:
sin(theta) = 0.1225.

To find the angle theta, we take the inverse sine (arcsin) of 0.1225:
theta = arcsin(0.1225).

Using a calculator, the correct value for theta is approximately 7.04 degrees.

Therefore, the correct answer is theta = 7.04 degrees.

So, after reviewing your calculations, it seems that theta = 18.008 is not correct. The correct answer is theta = 7.04 degrees.