# Formula for trajectory also applicable to decelerating bodies?

• ionowattodo
In summary, the formula d=0.5(g)(t^2) only applies to objects in free fall under Earth's gravity. For other accelerating or decelerating objects, there are various formulas that can be used depending on the known variables. These formulas only apply when there is a constant acceleration or deceleration.
ionowattodo
The formula that is applied to falling objects with disregard to any exterior forces except gravity is : d=0.5(g)(t^2)

Can it also be applied to decelerating bodies, such as cars, with disregard to friction?
Such as a car with an initial velocity slowing down to 0 m/s over a distance of 50 meters.

No, that formula only works for a body in free fall, starting from rest, in Earth's gravity (or equivalent).

For other accelerating or decelerating objects, there are various formulas you can use:
$$d = v_i t + \frac{1}{2}a t^2$$
if you know initial speed and acceleration
$$d = v_f t - \frac{1}{2}a t^2$$
if you know final speed and acceleration
$$d = \frac{(v_f + v_i)t}{2}$$
if you know initial and final speeds and time
$$d = \frac{v_f^2 - v_i^2}{2a}$$
if you know initial and final speed and acceleration

Of course, all these are only applicable when the acceleration is constant.

Can't you substitute gravity for deceleration?
Basically if you turn a deceleration car to it going upwards, it's the same as a thrown ball except with different deceleration rates

Well, yes, it's the same, but the rate of acceleration/deceleration is not g for a car. The formulas I gave in my post work for any constant acceleration/deceleration. But the one you were originally asking about is what you get when the acceleration happens to be equal to 9.8 m/s^2 (and the initial speed is zero).

The formula for trajectory, d=0.5(g)(t^2), is specifically designed to calculate the distance traveled by a falling object under the influence of gravity. It takes into account the acceleration due to gravity (g) and the time (t) the object has been falling. Therefore, it may not be applicable to decelerating bodies such as cars, as they are not solely affected by gravity but also by other external forces such as friction and air resistance.

To accurately calculate the distance traveled by a decelerating body, we would need to consider these additional forces and incorporate them into the formula. A more appropriate formula for a decelerating body would be d=V0t + 0.5at^2, where V0 is the initial velocity, a is the deceleration rate, and t is the time.

In the example given, the distance traveled by the car would be calculated using this formula as d= (0 m/s)(t) + 0.5(-a)(t^2) = -0.5at^2. This formula takes into account the initial velocity of 0 m/s and the deceleration rate (a) over a distance of 50 meters.

In conclusion, while the formula for trajectory may be applicable to falling objects, it cannot be used to accurately calculate the distance traveled by decelerating bodies such as cars. A different formula that takes into account all the relevant forces would need to be used for such calculations.

## 1. What is the formula for calculating trajectory for a decelerating body?

The formula for calculating the trajectory of a decelerating body is the same as that for an accelerating body, which is y = y0 + v0t - 1/2at^2, where y is the final position, y0 is the initial position, v0 is the initial velocity, t is the time, and a is the acceleration.

## 2. Can the trajectory formula be used for different types of decelerating bodies?

Yes, the formula for trajectory is applicable to any type of decelerating body, whether it is a vehicle, a projectile, or any other object.

## 3. How is the formula for trajectory derived?

The formula for trajectory is derived from the equations of motion, specifically the equation for displacement x = x0 + v0t + 1/2at^2, by substituting y for x and considering the vertical direction of motion.

## 4. Is air resistance taken into account in the formula for trajectory of a decelerating body?

No, the formula for trajectory does not take into account factors such as air resistance. It is based on the assumption that the body is moving in a vacuum or in a medium with constant density.

## 5. What other factors can affect the trajectory of a decelerating body besides acceleration?

Besides acceleration, other factors that can affect the trajectory of a decelerating body include initial velocity, angle of launch, and external forces such as friction and air resistance. These factors can alter the shape and height of the trajectory, making it more complex to calculate.

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