Initial Conditions:
- Initial speed of the particle, u = 5 m/s
- Angle of projection, θ = 53°
- Acceleration due to gravity, g = 10 m/s²

Time at which velocity is 45° with horizontal:
To find the time at which the velocity of the particle will be 45° with the horizontal, we need to determine the angle at that time. Let's call this angle α.

Calculating α:
We know that the horizontal component of velocity remains constant throughout the motion. Therefore, the horizontal component of velocity at any time t is given by:

Vx = u * cos(θ)

Similarly, the vertical component of velocity changes due to the acceleration due to gravity. The vertical component of velocity at any time t is given by:

Vy = u * sin(θ) - g * t

Now, we can find the angle α at any time using the formula:

tan(α) = Vy / Vx

Substituting the values of Vy and Vx, we get:

tan(α) = (u * sin(θ) - g * t) / (u * cos(θ))

Simplifying the equation, we get:

tan(α) = tan(θ) - (g * t) / u

Since tan(θ) is known, we can solve this equation to find the value of t at which α is 45°.

Calculating the time:
tan(45°) = tan(θ) - (g * t) / u

Simplifying the equation, we get:

1 = tan(θ) - (g * t) / u

Rearranging the equation, we get:

t = (u / g) * (tan(θ) - 1)

Plugging in the given values, we have:

t = (5 / 10) * (tan(53°) - 1)

Using a calculator, we can find the value of tan(53°) and substitute it into the equation to find the time t.

Final Answer:
The time at which the velocity of the particle will be 45° with the horizontal can be calculated as explained above. By substituting the given values into the equation, we find the value of t.

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