Video Transcript
A body was projected vertically downwards from the top of a tower whose height is 80 meters. Given that it covered 35.9 meters during the first second of its motion, find the time taken to reach the ground rounded to the nearest two decimal places. Let the acceleration due to gravity 𝑔 equal 9.8 meters per square second.
We are told that a body is projected vertically downwards from the top of a tower. The height of the tower is 80 meters. We know that the acceleration of the body is 9.8 meters per square second. We will let the initial velocity be 𝑢. We are also told that the body covers 35.9 meters during the first second. We can now solve this problem using the equations of motion or SUVAT equations. By considering the motion in the first second, we can see that the displacement 𝑠 is 35.9, the acceleration is 9.8, and the time is one.
We can calculate the value of 𝑢, the initial velocity, using the equation 𝑠 is equal to 𝑢𝑡 plus a half 𝑎𝑡 squared. Substituting in our values gives us 35.9 is equal to 𝑢 multiplied by one plus a half multiplied by 9.8 multiplied by one squared. The right-hand side simplifies to 𝑢 plus 4.9. Subtracting 4.9 from both sides of this equation gives us 𝑢 is equal to 31. This means that the initial velocity of the body is 31 meters per second.
We can now use this value to help us calculate the time taken to reach the ground. When the body hits the ground, the displacement is 80 meters as this is the height of the tower. We now know that 𝑢 is 31 and 𝑎 is still 9.8. Substituting these values into the same equation gives us 80 is equal to 31𝑡 plus a half multiplied by 9.8 multiplied by 𝑡 squared. The last term simplifies to 4.9𝑡 squared, and we have a quadratic equation.
Subtracting 80 from both sides gives us 4.9𝑡 squared plus 31𝑡 minus 80 is equal to zero. We can solve this quadratic equation using the quadratic formula to find two solutions for 𝑡. Our value of 𝑎 is 4.9, 𝑏 is 31, and 𝑐 is negative 80. Our two values of 𝑡 are 1.968 and so on and negative 8.294 and so on. We know that time can’t be negative, so we can rule out the second answer. We are asked to give the time correct to two decimal places. The time taken for the body to reach the ground is 1.97 seconds.
