Throw a rock horizontally from a cliff 50 m high. The rock lands 80 m away from the thrower. Find the initial velocity of the rock. [2 points]
Answer (1)
The time it takes for the rock to fall is t = g 2 h .
The horizontal distance is given by d = v 0 t .
Solve for v 0 : v 0 = d 2 h g .
Plug in the values: v 0 = 80 2 × 50 9.8 ≈ 25.04 m/s. The initial velocity of the rock is 25.04 m/s .
Explanation
Problem Analysis We are given that a rock is thrown horizontally from a cliff of height h = 50 m. The rock lands a horizontal distance d = 80 m away from the thrower. We need to find the initial horizontal velocity v 0 of the rock.
Finding the Time of Flight The time it takes for the rock to fall is determined by the height of the cliff and the acceleration due to gravity, g = 9.8 m/s 2 . The vertical motion is described by the equation h = 2 1 g t 2 . We solve for t :
t = g 2 h
Relating Distance, Velocity, and Time The horizontal distance is given by d = v 0 t . We substitute the expression for t into the equation for d :
d = v 0 g 2 h
Solving for Initial Velocity Now we solve for v 0 :
v 0 = g 2 h d = d 2 h g
Calculating the Initial Velocity We plug in the values for d , h , and g to find v 0 :
v 0 = 80 2 × 50 9.8 = 80 100 9.8 = 80 0.098 ≈ 80 × 0.313 = 25.04 . Therefore, the initial velocity of the rock is approximately 25.04 m/s.
Final Answer The initial velocity of the rock is approximately 25.04 m/s.
Examples
Understanding projectile motion is crucial in various real-world scenarios, such as in sports like baseball or golf, where the initial velocity and angle of launch determine the range of the ball. Similarly, in engineering, calculating the trajectory of projectiles is essential in designing systems that involve launching objects, like delivery drones or even in understanding the path of water in sprinkler systems. By applying the principles of physics and mathematics, we can accurately predict and control the motion of objects in flight.
