The speed of a bullet as it travels down the barrel of a...

The speed of a bullet as it travels down the barrel of a rifle toward the opening is given by $$v=\left(-5.00 \times 10^{7}\right) t^{2}+\left(3.00 \times 10^{5}\right) t$$ where $v$ is in meters per second and $t$ is in seconds. The acceleration of the bullet just as it leaves the barrel is zero. (a) Determine the acceleration and position of the bullet as functions of time when the bullet is in the barrel. (b) Determine the time interval over which the bullet is accelerated. (c) Find the speed at which the bullet leaves the barrel. (d) What is the length of the barrel?

The speed of a bullet as it travels down the barrel of a rifle toward the opening is given by
$$v=\left(-5.00 \times 10^{7}\right) t^{2}+\left(3.00 \times 10^{5}\right) t$$
where $v$ is in meters per second and $t$ is in seconds. The acceleration of the bullet just as it leaves the barrel is zero. (a) Determine the acceleration and position of the bullet as functions of time when the bullet is in the barrel. (b) Determine the time interval over which the bullet is
accelerated. (c) Find the speed at which the bullet leaves the barrel. (d) What is the length of the barrel?

Key Concepts

Kinematic Equations

Kinematics involves the equations that relate displacement, velocity, and acceleration of an object. In problems like this, knowing one kinematic function allows you to derive others through differentiation or integration, which is fundamental in understanding motion in one dimension.

Velocity and its Derivatives

The velocity function describes how the speed of an object changes over time. Differentiating the velocity function with respect to time gives the acceleration, which quantifies the rate of change of velocity, while integrating the velocity function gives the position, or displacement, of the object over time.

Differentiation to Find Acceleration

Acceleration is exactly the derivative of the velocity function. Calculus is employed to differentiate the given velocity expression with respect to time, yielding a function that tells us how quickly the bullet's speed is changing as it travels down the barrel.

Integration for Position Calculation

Since position is the integral of the velocity function, applying integration (and using appropriate initial conditions) will determine the bullet’s displacement or the distance traveled while it is being accelerated within the barrel.

Identifying Extremum Conditions

In this context, setting the acceleration function equal to zero allows you to find the time at which the velocity is at an extremum (in this case, maximum). This is critical in determining when the bullet is no longer accelerating and is ready to exit the barrel.

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