The velocity, \(v(t)\), in \(\mathrm{m} \mathrm{s}^{-1}\), of a projectile is given by $$ v(t)=10 \mathrm{e}^{-t} \quad t \geq 0 $$ (a) Calculate the distance travelled in the first 3 seconds. (b) Calculate the average speed over the first 3 seconds.

When exploring the concept of projectile motion, understanding the velocity-time relationship is crucial. Velocity, often denoted as v(t), represents the speed of an object in a specific direction at a particular time t. For a projectile, this relationship can describe how fast it's moving and in which direction at any point during its flight.

Specifically, in our exercise, the velocity function given by v(t) = 10e^-t is an expression that describes the projectile's velocity at any time t. As time progresses, the velocity changes according to this function. The negative exponent indicates that as time increases, the projectile's velocity decreases exponentially, which is a common scenario in physics when accounting for factors like air resistance or gravity that decelerate the projectile.

The definite integral is an indispensable tool in calculus, particularly when it comes to computing the area under a curve. In physical problems, like projectile motion, it can represent the total distance travelled by an object when the velocity function over time is known.

In the given problem, we compute the integral of the velocity function v(t) from time 0 to 3 seconds. This particular integral t_{0}^{3} v(t) dt = t_{0}^{3} 10e^{-t} dt represents the area under the velocity-time graph over the interval from 0 to 3, thereby giving us the distance the projectile has travelled during that time. The process involves finding the anti-derivative of the velocity function and then evaluating this at the upper and lower bounds of the interval.

Average speed is a measure of the total distance travelled divided by the total time taken. It is a straightforward calculation but a powerful indicator of overall velocity during a trip or motion sequence without regard for direction changes or fluctuations in speed.

In the context of projectile motion from our exercise, the formula Average Speed = Total Distance Travelled / Total Time allows us to summarize the entire trip's motion characteristics. With the projectile's distance travelled worked out using the definite integral, and the time known (3 seconds in our case), we easily plug in these values to acquire the average speed over that duration. This calculation gives us a single number that encapsulates the overall pace of the projectile throughout the given time frame.

Exponential functions, such as the one in our projectile motion problem, are fundamental in describing processes involving growth and decay. The general form of an exponential function is f(t) = ab^t, where a is a constant, b is the base, and t is the exponent.

In v(t) = 10e^-t, the base is Euler's number e (approximately 2.71828), a constant that frequently appears in growth and decay problems, while the exponent is -t. This negative exponent signals a decay process, meaning the projectile's velocity decreases over time. Exponential functions are especially good at modeling scenarios where changes occur relative to the current value, such as chilling of a warm object in air or, indeed, the decreasing velocity of a projectile subjected to air resistance.