1. Física
Matemática

2. tt0 t
s(t0)
s(t)
s
Vector posición
𝑆(t) = 𝑥 𝑡 + 𝑦 𝑡
𝑆 𝑡 = 𝑖 𝑥 𝑡 + 𝑗 𝑦(𝑡)
𝑆(𝑡 + ∆𝑡) = 𝑆(𝑡) + ∆𝑠AB
∆𝑠AB= 𝑆(𝑡 + ∆𝑡) − 𝑆 𝑡

3. 𝑉 𝑡 =
𝑑 𝑆(𝑡)
𝑑𝑡
=
𝑑 𝑥(𝑡)
𝑑𝑡
+
𝑑𝑦(𝑡)
𝑑𝑡
=
𝑖 𝑑𝑥(𝑡)
𝑑𝑦
+
𝑗𝑑𝑦(𝑡)
𝑑𝑡
= 𝑖 𝑉x(t) + 𝑗 𝑉y(t)
𝑉 𝑡 = 𝑉x(t)+𝑉y(t)
𝑉 𝑡 =
𝑑 𝑥(𝑡)
𝑑𝑡
+
𝑑𝑦(𝑡)
𝑑𝑡
tt0 t
s(t0)
s(t)
s
Vector velocidad

4. 𝑗 𝑎 𝑦 𝑡 = 𝑎 𝑦 𝑡 =
𝑑𝑣 𝑦(𝑡)
𝑑𝑡
= 𝑗
𝑑𝑣 𝑦
𝑑𝑡
⟹ 𝑗 𝑎 𝑦 𝑡 = 𝑗
𝑑𝑣 𝑦(𝑡)
𝑑𝑡
Tiro oblicuo
𝑎 𝑥 𝑡 + 𝑎 𝑦 𝑡 = 𝑎 𝑡 = 0𝑖 + 𝑗 . 𝑔 𝑡 = 𝑗 . (−9,8 𝑚
𝑠𝑒𝑔2)
𝑖 𝑎 𝑥 𝑡 = 𝑎 𝑥 𝑡 =
𝑑 𝑣 𝑥(𝑡)
𝑑𝑡
= 𝑖
𝑑𝑣 𝑥
𝑑𝑡
⟹ 𝑖 𝑎 𝑥 𝑡 = 𝑖
𝑑𝑣 𝑥(𝑡)
𝑑𝑡
𝑑𝑣 𝑦 𝑡 = −9,8 𝑑𝑡
𝑎 𝑥 𝑡 =
𝑑𝑣 𝑥(𝑡)
𝑑𝑡
⟹ 0 =
𝑑𝑣 𝑥(𝑡)
𝑑𝑡
𝑑𝑣 𝑥 𝑡 = 0 𝑑𝑡
𝑣 𝑥 𝑡 = 𝑐 (1)
𝑎 𝑦 𝑡 =
𝑑𝑦(𝑡)
𝑑𝑡
⟹ −9,8 =
𝑑𝑣 𝑦(𝑡)
𝑑𝑡
𝑣 𝑦 𝑡 = −9,8 + 𝑐 (2)

5. Aceleración en el eje x
Para 𝑡 = 𝑡0
𝑣 𝑥 𝑡 = 𝑣 𝑥 𝑡0 = 𝑣 𝑥0
𝑣 𝑥 𝑡0 = 𝑐
∴ 𝑣 𝑥 𝑡 = 𝑣 𝑥 𝑡 𝑜 = 𝑐𝑡𝑒
𝑣 𝑥 𝑡0
𝑣 𝑥 𝑡
𝑑𝑣 𝑥 𝑡 = 𝑡0
𝑡
0 𝑑𝑡
𝑣 𝑥 𝑡 − 𝑣 𝑥 𝑡0 = 0 ⟹ 𝑣 𝑥 𝑡 = 𝑣 𝑥(𝑡0)
Aceleración en el eje y
𝑑𝑣 𝑦
𝑑𝑡
= 𝑎𝑦 = −𝑔
𝑣 𝑦 𝑡 = 𝑣 𝑦 𝑡0 − 𝑔𝑡
𝑑𝑣 𝑦 𝑡 = −𝑔 𝑑𝑡 ⇒ 𝑣 𝑦 = −𝑔𝑡 + 𝑐
𝑣 𝑦 𝑡0
𝑣 𝑦 𝑡
𝑑𝑣 𝑦 𝑡 = 𝑡0
𝑡
−𝑔𝑑𝑡 ⇒ 𝑣 𝑦 = 𝑣 𝑦 𝑡0 − 𝑔( 𝑡 −

6. 𝑥 𝑡0
𝑥 𝑡
𝑑𝑥 = 𝑡0
𝑡
𝑣 𝑥 𝑡 𝑑𝑡
𝑖𝑥 𝑡 = 𝑖 𝑥 𝑡0 + 𝑖𝑣 𝑥 𝑡0 𝑡
𝑥 𝑡 = 𝑥 𝑡0 + 𝑣 𝑥 𝑡 𝑡

7. Si 𝑡0 = 0
𝑣 𝑦 𝑡 = 𝑣 𝑦 𝑡0 − 𝑔𝑡
𝑣 𝑦 𝑡 = 𝑗𝑣 𝑦 𝑡 = 𝑗 𝑣 𝑦 𝑡 − 𝑔 𝑡 − 𝑡0
𝑑𝑦 𝑡
𝑑𝑡
= 𝑣 𝑦 𝑡 ⇒
𝑑𝑗𝑦 𝑡
𝑑𝑡
= 𝑗 𝑣 𝑦 𝑡
𝑗
𝑑𝑦 𝑡
𝑑𝑡
= 𝑗 𝑣 𝑦 𝑡
𝑑𝑦 𝑡 = 𝑣 𝑦 𝑡
𝑑𝑦 𝑡 = 𝑣 𝑦 𝑡0 − 𝑔 𝑡 − 𝑡0 𝑑𝑡
𝑦 𝑡 = 𝑣 𝑦 𝑡0 𝑡 −
𝑔𝑡2
2
+ 𝑔𝑡0 𝑡 + 𝑐
Para 𝑡0 = 0 ⇒ 𝑦 𝑡 = 𝑦 𝑡0
𝑦 𝑡0 = 𝑣 𝑦 𝑡0 𝑡0 −
𝑔 𝑡0
2
2
+ 𝑔𝑡0 𝑡0 + 𝑐
𝑦 𝑡0
𝑦 𝑡
𝑑𝑦 = 𝑡0
𝑡
𝑣 𝑦 𝑡0 − 𝑔 𝑡 − 𝑡0 𝑑𝑡
𝑦 𝑡 𝑦 𝑡
𝑦 𝑡0
= 𝑣 𝑦 𝑡0 𝑡 − 𝑡 𝑡
𝑡0
−
𝑔𝑡2
2
𝑡
𝑡0
+ 𝑔𝑡0
𝑡
𝑡0
𝑦 𝑡 − 𝑦 𝑡0 = 𝑣 𝑦 𝑡0 𝑡 − 𝑡0 −
𝑔𝑡2
2
+
𝑔 𝑡0
2
2
+
𝑔 𝑡0 − 𝑔 𝑡0
2
𝑦 𝑡 − 𝑦 𝑡0 = − 𝑦 𝑡0 𝑡 − 𝑡0 −
𝑔𝑡2
2
−
𝑔 𝑡0
2
2
+ 𝑔 𝑡0 − 𝑡 − 𝑡0
2
𝑦 𝑡 = 𝑦 𝑡0 + 𝑣 𝑦 𝑡0 𝑡 − 𝑡0 −
1
2
𝑔 𝑡 − 𝑡0
2
Si para 𝑡0 = 0 𝑦 𝑡0 = 0
𝑦 𝑡 = 𝑦0 + 𝑣 𝑦𝑜 𝑡 −
𝑔𝑡2
2