A target lies flat on the ground 8 m from the side of a cliff that is 20 m tall. A physics student rolls a ball off the horizontal cliff in the direction of the target. Air resistance is negligible. Determine the horizontal speed with which the ball must leave the top of the cliff if it is to strike the target.
how long does it take to fall 20m? 4.9t^2 = 20
Now, use that t to find the horizontal speed, since
speed = distance/time
To determine the horizontal speed with which the ball must leave the top of the cliff, we can consider the following steps:
Step 1: Determine the time taken for the ball to fall from the cliff to the ground.
Given that the height of the cliff is 20 m, we can use the kinematic equation:
h = (1/2) * g * t^2
where h is the height of the cliff, g is the acceleration due to gravity (9.8 m/s^2), and t is the time taken.
Plugging in the values, we have:
20 = (1/2) * 9.8 * t^2
Simplifying the equation, we get:
t^2 = 20 * 2 / 9.8
t^2 ≈ 4.08
t ≈ √4.08
t ≈ 2.02 seconds
Step 2: Determine the horizontal distance traveled by the ball during the time it takes to fall.
Since the horizontal distance is given as 8 m, and there is no horizontal acceleration, we can use the equation:
d = v * t
where d is the horizontal distance, v is the horizontal velocity, and t is the time taken.
Plugging in the values, we have:
8 = v * 2.02
Simplifying the equation, we get:
v ≈ 8 / 2.02
v ≈ 3.96 m/s
Therefore, the horizontal speed with which the ball must leave the top of the cliff in order to strike the target is approximately 3.96 m/s.
To determine the horizontal speed with which the ball must leave the top of the cliff to strike the target, we can use the principles of projectile motion. Here's how you can solve it step by step:
Step 1: Analyze the given information.
- The distance from the target to the cliff is 8 m.
- The height of the cliff is 20 m.
- Neglect air resistance.
Step 2: Find the time of flight.
- The ball will fall vertically downwards under the influence of gravity, while also moving horizontally towards the target.
- The time it takes for the ball to hit the ground can be found using the vertical motion formula: h = (1/2)gt^2, where h is the height of the cliff (20 m), g is the acceleration due to gravity (9.8 m/s^2), and t is the time of flight.
- Rearranging the formula to solve for t gives: t = √(2h/g).
- Plugging in the given values, we get: t = √(2*20/9.8) ≈ 2.02 seconds.
Step 3: Determine the horizontal distance traveled.
- Since the horizontal speed remains constant throughout the motion, we can calculate the horizontal distance traveled using the formula: d = v*t, where d is the horizontal distance, v is the horizontal velocity, and t is the time of flight.
- The horizontal distance traveled is equal to the distance from the cliff to the target, which is 8 m.
Step 4: Calculate the horizontal velocity.
- Rearrange the formula to solve for v: v = d/t.
- Plugging in the values, we get: v = 8 m / 2.02 s ≈ 3.96 m/s.
Therefore, the ball must leave the top of the cliff with a horizontal speed of approximately 3.96 m/s to strike the target.