A projectile is fired vertically upward with an initial...

A projectile is fired vertically upward with an initial velocity of $49 \mathrm{m} / \mathrm{s}$ from a platform $150 \mathrm{m}$ high. a) How long will it take the projectile to reach its maximum height? b) What is the maximum height? c) How long will it take the projectile to pass its starting point on the way down? d) What is the velocity when it passes the starting point on the way down? e) How long will it take the projectile to hit the ground? f) What will its speed be at impact?

A projectile is fired vertically upward with an initial velocity of $49 \mathrm{m} / \mathrm{s}$ from a platform $150 \mathrm{m}$ high.
a) How long will it take the projectile to reach its maximum height?
b) What is the maximum height?
c) How long will it take the projectile to pass its starting point on the way down?
d) What is the velocity when it passes the starting point on the way down?
e) How long will it take the projectile to hit the ground?
f) What will its speed be at impact?

Key Concepts

Maximum Height

The maximum height in a vertical projectile problem is reached when the upward velocity of the projectile becomes zero. This concept is critical because it marks the turning point of the projectile's trajectory and allows the use of kinematic equations to compute both the time to reach this point and the corresponding displacement from the launch position.

Solving Quadratic Equations

In problems involving projectile motion with an initial height, the displacement equation becomes a quadratic equation in time. Solving this quadratic equation is necessary to determine specific times during the motion, such as when the projectile returns to a particular height or hits the ground.

Projectile Motion

Projectile motion refers to the motion of an object that is subject only to the acceleration of gravity (neglecting air resistance). In vertical projectile problems, the motion is one-dimensional with a constant downward acceleration, and the same set of kinematic equations can be used to predict the position, velocity, and time at various points during the flight.

Kinematic Equations

The kinematic equations are fundamental formulas that relate displacement, initial velocity, final velocity, acceleration, and time for objects moving with constant acceleration. These equations are essential for analyzing projectile motion, such as finding the time to reach maximum height, determining the displacement at any given time, and calculating the velocity during upward or downward motion.

Impact Speed Calculation

The impact speed is determined by calculating the velocity of the projectile just before it strikes the ground. It involves using kinematic relationships that take into account the entire displacement from the launch point to the ground, and it is important to distinguish between velocity (which includes direction) and speed (the magnitude of velocity).

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