# Solved: Relativistic Speed: Force of 30N Reaches 99% c in 7 x 10⁴ Sec

• neelakash
In summary: Or is it just in the relativistic frame?In summary, the increase in mass is only in the relativistic frame.
neelakash

## Homework Statement

A particle of mass 1 gm starts from rest and moves under the action of a force of 30
Newtons defined in the rest frame. It will reach 99% the velocity of light in time
(a) 9.9 x 10³ sec (b) 7 x 10⁴ sec (c) 0.999 sec (d) 0.7 sec

## The Attempt at a Solution

I can safely disregard (c) and (d)

(a) can be found by kinematic calculation.But as the particle moves in relativistic speed, this classical calculation does not hold.

(b) looks correct to me.

But how to prove this?

Can anyone suggest anything?

as speed increases,the value of m(v) also increases,thus reducing the acceleration.Thus, it takes longer time...In that respect (b) looks plausible.

neelakash said:

## Homework Statement

A particle of mass 1 gm starts from rest and moves under the action of a force of 30
Newtons defined in the rest frame. It will reach 99% the velocity of light in time
(a) 9.9 x 10³ sec (b) 7 x 10⁴ sec (c) 0.999 sec (d) 0.7 sec

## The Attempt at a Solution

I can safely disregard (c) and (d)

(a) can be found by kinematic calculation.But as the particle moves in relativistic speed, this classical calculation does not hold.

(b) looks correct to me.

But how to prove this?

Can anyone suggest anything?

In relativity, $$F = \frac{dp}{dt}$$ still holds. For a constant force you have then

$$p_{final} - p_{initial} = F \Delta t$$

setting p initial equal to zero and using for the final $$p= \gamma m v$$ gives the correct answer, which is indeed b.

Note: your argument is correct and it's sufficient if all you need is to pick the correct answer quickly from a choice of answers. I just wanted to show you how to do the calculation if you needed that at some point.

Last edited:

Tell me one thing.The force is specified to be 30 dyne in its own rest frame.So, is the increase of mass is to be considered at all?

## 1. How is relativistic speed defined?

Relativistic speed is the speed at which an object is moving relative to an observer, taking into account the effects of special relativity.

## 2. What is the significance of reaching 99% of the speed of light?

Reaching 99% of the speed of light is significant because it is very close to the maximum possible speed in the universe, and at this speed, the effects of special relativity become more prominent.

## 3. How is the force of 30N calculated in this problem?

The force of 30N is calculated using the equation F = ma, where m is the mass of the object and a is its acceleration. In this case, the given acceleration is the rate at which the object's speed is increasing over time.

## 4. What is the time interval of 7 x 10⁴ seconds in this problem?

The time interval of 7 x 10⁴ seconds represents the duration of the object's acceleration, or the time it takes for the object to reach 99% of the speed of light from its initial speed.

## 5. How does special relativity affect the force required to reach relativistic speeds?

According to special relativity, as an object approaches the speed of light, its mass increases and therefore requires more force to accelerate. This means that as an object reaches higher speeds, more force is needed to continue accelerating it.

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