An arrow is shot at a 30.0° angle with the horizontal. It has a velocity of 49 m/s.
(a) How high will the arrow go?
m
(b) What horizontal distance will it travel?
m
24
To determine the answers to these questions, we can utilize the physics principles of projectile motion. Projectile motion deals with the motion of objects that are thrown or projected into the air, under the influence of gravity.
(a) To find how high the arrow will go, we need to determine the maximum height reached by the projectile. The key parameter for this calculation is the vertical component of the arrow's velocity.
Step 1: Find the vertical component of the velocity.
In projectile motion, the vertical component of velocity (Vy) can be calculated using the equation:
Vy = V * sin(θ)
where V is the magnitude of the velocity (49 m/s) and θ is the launch angle (30.0°).
Vy = 49 m/s * sin(30.0°)
Vy = 49 m/s * 0.5
Vy = 24.5 m/s
Step 2: Calculate the time of flight.
The time it takes for the projectile to reach its maximum height and then return to the same height is called the time of flight (T). The formula to calculate T is:
T = 2 * (Vy / g)
where g is the acceleration due to gravity (9.8 m/s²).
T = 2 * (24.5 m/s / 9.8 m/s²)
T = 4.99 s
Step 3: Find the maximum height.
The maximum height of the arrow can be determined using the formula:
H = (Vy²) / (2 * g)
H = (24.5 m/s)² / (2 * 9.8 m/s²)
H = 598.02 m²/s² / 19.6 m²/s²
H = 30.5 m
Therefore, the arrow will go up to a height of 30.5 meters.
(b) To find the horizontal distance traveled by the arrow, we need to determine the horizontal component of the velocity.
Step 1: Find the horizontal component of the velocity.
The horizontal component of velocity (Vx) can be calculated using the equation:
Vx = V * cos(θ)
where V is the magnitude of the velocity (49 m/s) and θ is the launch angle (30.0°).
Vx = 49 m/s * cos(30.0°)
Vx = 49 m/s * 0.866
Vx = 42.41 m/s
Step 2: Calculate the horizontal distance.
The horizontal distance traveled (D) can be calculated using the formula:
D = Vx * T
where Vx is the horizontal component of the velocity and T is the time of flight.
D = 42.41 m/s * 4.99 s
D = 211.85 m
Therefore, the arrow will travel a horizontal distance of 211.85 meters.