Question

An arrow is shot horizontally with a speed of \(20 . \mathrm{m} / \mathrm{s}\) from the top of a tower \(60 . \mathrm{m}\) high. The time to reach the ground will be a) \(8.9 \mathrm{~s}\). b) \(7.1 \mathrm{~s}\). c) \(3.5 \mathrm{~s}\) d) \(2.6 \mathrm{~s}\). e) \(1.0 \mathrm{~s}\)

Short answer

Choose the correct answer. a) 2.5 s b) 3.0 s c) 3.5 s d) 4.0 s Answer: c) 3.5 s

Step-by-step solution

1. Write down the initial conditions

We first extract the given information from the problem statement: Initial vertical velocity \(v_0 = 0\) m/s (since it's a horizontal shot) Height of the tower \(h = 60\) m Acceleration due to gravity \(a = -9.8\) m/s² (negative sign indicates downward acceleration)

2. Employ the kinematic equation for the vertical motion

We use the equation relating displacement, initial velocity, acceleration, and time: \(h = v_0t + \frac{1}{2}at^2\) Plug in the given values: \(60 = 0 + \frac{1}{2}(-9.8)t^2\)

3. Solve for the time t

Solve the equation for t: \(60 = -4.9t^2\) \(t^2 = \frac{60}{4.9}\) \(t^2 = 12.24\) \(t = \sqrt{12.24}\) \(t \approx 3.5\) s

4. Choose the correct option

Now, compare the calculated time with the options given in the problem statement. The time it takes for the arrow to reach the ground after being shot horizontally from the top of the tower is approximately 3.5 s. The correct answer is: c) \(3.5 \mathrm{~s}\).

Key concepts

Projectile Motion

Understanding projectile motion is essential for analyzing the flight of objects, like arrows or balls, after being launched. When an object is projected with an initial velocity and allowed to move under the influence of gravity, it follows a curved path. This path is a combination of horizontal and vertical motions, where the horizontal motion occurs at a constant velocity, and the vertical motion is subject to acceleration due to gravity.

Consider our example of the arrow shot horizontally from the tower. The arrow's horizontal motion stays constant because there is no acceleration acting in that direction (assuming we neglect air resistance for simplicity). The only force at play is gravity, which affects the arrow's vertical motion. Therefore, while the arrow moves forward horizontally, it also accelerates downward, creating a parabolic trajectory typical of projectile motion.

Acceleration due to Gravity

The term acceleration due to gravity refers to the acceleration experienced by an object when gravity is the only force acting upon it. On Earth, this acceleration has a standard value of approximately 9.8 m/s², directed downward towards the center of the Earth. This constant acceleration impacts the vertical component of projectile motion.

In our textbook problem, gravity compels the arrow to accelerate downwards at this rate, which is why we use the value in the kinematic equation to determine how long it takes for the arrow to reach the ground. It's critical to note that the acceleration due to gravity is a vector quantity, thus it's often represented with a negative sign when solving for motion in the downward direction.

Free Fall Motion

When an object is only under the influence of gravity, without any initial vertical velocity, the motion is known as free fall motion. In the context of our arrow problem, the arrow experiences free fall in the vertical direction because it was shot horizontally, implying it had an initial vertical velocity of zero.

An object in free fall will have its vertical velocity increase by 9.8 m/s for every second it falls due to the acceleration due to gravity. This factor is key in predicting how long it takes an object to hit the ground from a given height, which is precisely what we calculated using the kinematic equation for the arrow’s descent from the tower.

Initial Velocity

The concept of initial velocity is particularly important when dealing with any form of motion. It represents the velocity of an object before any forces or accelerations are applied to it. In projectile motion, initial velocity can be broken down into horizontal and vertical components. For the arrow shot from the tower, the initial horizontal velocity is significant, at 20 m/s, while the initial vertical velocity is 0 m/s, since it's not given any initial push upwards or downwards.

It's essential to recognize that the initial vertical velocity plays a crucial role in determining the object's trajectory and time of flight. However, in the instance of zero initial vertical velocity, we can calculate the time taken for the arrow to reach the ground by focusing solely on free fall motion due to gravity, just as we did in solving the textbook exercise.