A frog leaps at [tex]$8.40 m / s$[/tex] at a [tex]$58.7^{\circ}$[/tex] angle to the horizon.
How far away does the frog land?
[tex]$\Delta x=[?] m$[/tex]
Answer (1)
Calculate the horizontal and vertical components of the initial velocity: v 0 x = v 0 cos ( θ ) and v 0 y = v 0 sin ( θ ) .
Determine the time to reach maximum height: t = g v 0 y .
Calculate the total time of flight: T = 2 t .
Calculate the horizontal distance: Δ x = v 0 x × T . The horizontal distance is 6.392 m .
Explanation
Problem Analysis We are given the initial speed v 0 = 8.40 m / s and the launch angle θ = 58. 7 ∘ . We want to find the horizontal distance Δ x the frog travels before landing.
Calculate Velocity Components First, we need to find the horizontal and vertical components of the initial velocity. We can use trigonometry for this:
Horizontal component: v 0 x = v 0 cos ( θ ) Vertical component: v 0 y = v 0 sin ( θ )
Time to Reach Max Height Next, we calculate the time it takes for the frog to reach its maximum height. At the maximum height, the vertical velocity is zero. We use the following kinematic equation:
v y = v 0 y − g t = 0
where g = 9.8 m / s 2 is the acceleration due to gravity. Solving for t :
t = g v 0 y
Total Time of Flight The total time of flight, T , is twice the time to reach the maximum height:
T = 2 t
Calculate Horizontal Distance Finally, we calculate the horizontal distance traveled using the formula:
Δ x = v 0 x × T
Now, let's plug in the values and calculate the result.
v 0 x = 8.40 × cos ( 58. 7 ∘ ) = 8.40 × 0.5204 = 4.371 m / s v 0 y = 8.40 × sin ( 58. 7 ∘ ) = 8.40 × 0.8545 = 7.178 m / s t = 9.8 7.178 = 0.732 s T = 2 × 0.732 = 1.464 s Δ x = 4.371 × 1.464 = 6.392 m
Final Answer Therefore, the horizontal distance the frog travels is approximately 6.392 m .
Examples
Understanding projectile motion, like the frog's leap, is crucial in sports such as basketball or soccer, where players aim to throw or kick a ball to a specific location. By calculating the initial velocity and launch angle, athletes can optimize their performance. Similarly, engineers use these principles to design systems ranging from launching satellites into orbit to predicting the trajectory of water in irrigation systems, ensuring efficient and accurate delivery.
