A projectile is launched at an angle of 20.0° above the horizontal. What is its initial speed if it hits a target that is located a horizontal distance of 175 m from the launch point and 10.5 m above the launch level?
To find the initial speed of the projectile, we can use the equations of motion for projectile motion. Given the launch angle, horizontal distance, and vertical displacement, we can break down the projectile's motion into horizontal and vertical components.
Let's consider the horizontal motion first. The horizontal velocity remains constant throughout the projectile's flight because there is no horizontal acceleration. Thus, the horizontal displacement is given by:
horizontal displacement = horizontal velocity × time
Since we know the horizontal displacement is 175 m and the launch angle is 20.0°, we can find the horizontal velocity using trigonometry. The horizontal velocity can be expressed as:
horizontal velocity = initial speed × cos(angle)
Similarly, let's consider the vertical motion. The vertical displacement can be determined as:
vertical displacement = (vertical velocity × time) + (0.5 × acceleration × time^2)
The initial vertical velocity can be found using trigonometry as well:
vertical velocity = initial speed × sin(angle)
Since the projectile lands at the same vertical level (10.5 m), the vertical displacement is zero. Therefore, we can set up the equation:
0 = (vertical velocity × time) + (0.5 × acceleration × time^2)
We know the acceleration due to gravity is -9.8 m/s^2, as the object is moving upward against gravity. By substituting the values, we can solve for time.
To find the total flight time, we can use the horizontal motion equation:
horizontal displacement = horizontal velocity × time
Substituting the values, we can find the value of time.
With the value of time and the initial speed equation, we can solve for the initial speed.
Let me calculate it for you.
S is initial speed
u = S cos 20 forever
so
175 = S cos 20 * t
so t = 175 / (S cos 20)
then the height
10.5 = 0 + S sin 20 * t - 4.9 t^2
10.5 = 175 tan 20 - 4.9 [175^2 / (S^2 cos^2 20)] solve for S