A frog jumps from the ground at an angle of 25.0° to the horizontal and at an initial velocity of 0.6 m/s. What is the maximum height?
Jaja
To find the maximum height reached by the frog, we can use the concept of projectile motion. The motion of the frog can be separated into two independent components: horizontal motion and vertical motion.
First, let's analyze the vertical motion. The initial velocity can be broken down into two components - one in the horizontal direction (0.6 m/s * cos(25°)) and one in the vertical direction (0.6 m/s * sin(25°)).
The vertical motion of the frog can be described by the equation:
final vertical velocity (Vf) = initial vertical velocity (Vi) + (acceleration due to gravity * time)
At the maximum height, the final vertical velocity will be zero, as the frog reaches its highest point and starts to fall back down. The acceleration due to gravity is approximately 9.8 m/s² and acts downward. Thus, we can rearrange the equation to solve for the time taken to reach maximum height:
0 = (0.6 m/s * sin(25°)) + (-9.8 m/s² * t)
0.6 m/s * sin(25°) = 9.8 m/s² * t
The sine of 25° can be calculated as approximately 0.4226. Substituting this value, we can solve for t:
0.6 m/s * 0.4226 = 9.8 m/s² * t
t = (0.6 * 0.4226) / 9.8
Calculating this expression gives us t ≈ 0.0259 seconds. This is the time taken to reach maximum height.
Now, let's calculate the maximum height using the formula:
maximum height (h) = initial vertical velocity (Vi) * time (t) + (1/2) * (acceleration due to gravity) * (time)^2
Plugging in the given values:
h = (0.6 m/s * sin(25°) * 0.0259 s) + (1/2) * (-9.8 m/s²) * (0.0259 s)^2
Evaluating this expression, the maximum height would be approximately 0.00826 meters or 8.26 centimeters.
To find the maximum height reached by the frog, we can use the projectile motion equations.
Given:
Initial velocity (u) = 0.6 m/s
Launch angle (θ) = 25.0°
Step 1: Resolve the initial velocity into its horizontal and vertical components.
The horizontal component (ux) can be found using the equation:
ux = u * cos(θ)
Substituting the given values:
ux = 0.6 m/s * cos(25.0°) = 0.541 m/s
The vertical component (uy) can be found using the equation:
uy = u * sin(θ)
Substituting the given values:
uy = 0.6 m/s * sin(25.0°) = 0.255 m/s
Step 2: Determine the time taken to reach the maximum height.
The time taken to reach the maximum height (t) can be calculated using the following equation:
t = uy / g
where g is the acceleration due to gravity, which is approximately 9.8 m/s^2.
Substituting the values:
t = 0.255 m/s / 9.8 m/s^2 ≈ 0.026 seconds
Step 3: Calculate the maximum height.
The maximum height (h) can be found using the equation:
h = uy * t - 0.5 * g * t^2
Substituting the values:
h = 0.255 m/s * 0.026 s - 0.5 * 9.8 m/s^2 * (0.026 s)^2 ≈ 0.0034 meters
Therefore, the maximum height reached by the frog is approximately 0.0034 meters.