Question

Distance Traveled by a Projectile An object is shot straight upward from sea level with an initial velocity of 400 ft/sec. (a) Assuming gravity is the only force acting on the object, give an upper estimate for its velocity after 5 sec have elapsed. Use \(g=32 \mathrm{ft} / \mathrm{sec}^{2}\) for the gravitational constant. (b) Find a lower estimate for the height attained after 5 sec.

Short answer

The estimated final velocity after 5 seconds is : 240 ft/sec and the height attained after 5 seconds is: 1600ft

Step-by-step solution

Calculate Final Velocity after 5 seconds

Using the formula \(v = u - gt\), where \(v\) is final velocity, \(u\) is initial velocity, \(g\) is the gravitational constant, and \(t\) is time. Substitute the given values \(u = 400 ft/sec\), \(g = 32 ft/sec^2\), and \(t = 5 sec\) into the formula to find the final velocity. Here, the final velocity represents the change in speed due to the force of gravity acting on the object for a specified time frame. This step is important for estimating the speed of the object after the specified time has elapsed.

Calculate the Height attained after 5 seconds

Using the formula \(h = ut - 1/2 g t^2\), where \(h\) is the height, \(u\) is initial velocity, \(g\) is the gravitational constant, and \(t\) is time. Substitute the given values \(u = 400 ft/sec\), \(g = 32 ft/sec^2\), and \(t = 5 sec\) into the formula to find the height. The height represents the distance or the vertical displacement that the object has travelled in the specified time. This step is crucial for finding the distance travelled by the object after a specified time has lapsed.

Key concepts

Final Velocity Calculation

The concept of final velocity is crucial when delving into the realms of projectile motion calculus. It refers to the speed at which an object is moving at the end of a given time period, taking into account the influences that have acted upon it, such as gravity. In our exercise, an object is projected upwards with an initial velocity of 400 ft/sec. After 5 seconds, we want to calculate its final velocity.

To achieve this, we apply the formula:
\[ v = u - gt \]
where \( v \) represents the final velocity, \( u \) is the initial velocity, \( g \) is the acceleration due to gravity (32 ft/sec² in this case), and \( t \) is the time in seconds. Substituting the known values, we acquire the final velocity.

This process is imperative for predicting how fast the object will be moving after a certain time has elapsed, and it allows us to make educated estimates on the various aspects of the projectile’s journey.

Gravitational Acceleration

Gravity, the force that draws objects towards the center of the Earth, plays a pivotal role in determining the motion of projectiles. In our exercise, we refer to the gravitational acceleration as \( g \), and it has been provided as 32 ft/sec². This value is critical for calculating both the final velocity and the height estimation of the object in projectile motion.

Gravitational acceleration is constant near the Earth's surface, offering a predictable influence on the motion of objects. It causes the object's upward velocity to decrease by 32 ft/sec each second. Understanding this concept is essential because it allows us to describe the trajectory of an object in projectile motion accurately. It's pertinent to note that this acceleration value is a near-surface average and may differ based on altitude and geographic location.

Height Estimation of a Projectile

The height estimation of a projectile is a vital calculation for understanding how high an object travels when projected upwards. In the exercise, we find the height by employing the formula:
\[ h = ut - \frac{1}{2} gt^2 \]
Here, \( h \) is the height at any time \( t \), \( u \) is the initial velocity, and \( g \) is the gravitational acceleration. After substituting the values into the formula, we determine the maximum height the object reaches after 5 seconds.

Height estimation is fundamental for various applications, such as engineering and ballistics because it explains how vertical displacement changes over time under the influence of gravity. An object's height at any given moment during its flight is a kinetic snapshot of its journey against gravity's pull.